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PROJECTIVE  GEOMETRY 


MODERN  MATHEMATICAL  TEXTS 

EDITED   BY 

CHARLES  S.  SLIGHTER 


ELEMENTARY  MATHEMATICAL  ANALYSIS 

BY  CHARLES  S.  SLIGHTER 
490  pages,  5  x  7H,  Illustrated $2.50 

MATHEMATIC  FOR  AGRICULTURAL 

STUDENTS 
BY  HENRY  C.  WOLFF 
311  pages,  5  x  7}i,  Illustrated, $1.50 

CALCULUS 

BY  HERMAN  W.  MABCH  AND  HENRY  C.  WOLFF 
|   360  pages,  5  x  7ft,  Illustrated $2.00 

PROJECTIVE  GEOMETRY 

BY  L.  WAYLAND  DOWLINO 
316  pages,5  x  7^,  Illustrated $2.00 


MODERN    MATHEMATICAL    TEXTS 

EDITED   BY   CHARLES   S.    SLIGHTER 

PEOJECTIVE 
GEOMETRY 


BY 
L.  WAYLAND  BOWLING,  PH.  D. 

ASSOCIATE    PROFESSOR    OP    MATHEMATICS 
UNIVERSITY    OF    WISCONSIN 


FIRST  EDITION 


McGRAW-HILL  BOOK  COMPANY,  INC. 

239  WEST  39TH  STREET.     NEW  YORK 


LONDON:  HILL  PUBLISHING  CO.,  LTD. 
6  &  8  BOUVERIE  ST.,  E.  C. 

1917 


COPYRIGHT,  1917,  BY  THE 
MCGRAW-HILL  BOOK  Co.,  INC. 


THE  MAPLE  PRESS  YORK  PA 


Sciences 
L*rery 


PREFACE 

The  present  volume  embodies  a  course  of  lectures  on  Projective 
Geometry  given  by  the  author  for  a  number  of  years  at  the  Uni- 
versity of  Wisconsin.  The  synthetic  point  of  view  was  chosen 
primarily  to  develop  the  power  of  visualization  and  of  pure  geo- 
metric analysis  for  young  men  and  women  preparing  to  teach 
geometry  in  our  secondary  schools.  Such  a  course  should  naturally 
avoid  a  review  of  the  subject  matter  of  Elementary  Geometry  and, 
at  the  same  time,  should  not  be  so  far  removed  from  familiar  con- 
cepts as  to  lose  connection  with  them.  In  the  second  place,  the 
synthetic  treatment  of  loci  of  the  second  order  and  of  the  second 
class  opens  up  a  new  field  to  the  student  familiar  with  analytical 
processes  and  has  certain  advantages  in  arousing  his  enthusiasm 
for  continued  work  in  mathematics. 

No  especial  preparation  beyond  Elementary  Geometry  and  a 
slight  knowledge  of  Trigonometry  is  required  in  order  to  read  this 
book  with  perfect  understanding.  The  reader  who  knows  his 
Analytic  Geometry  will  often  find  himself  on  familiar  ground,  but 
no  knowledge  beyond  the  use  of  coordinate  axes  is  assumed. 

The  book  is  frankly  patterned  after  Reye's  Geometric  der  Lage, 
with  the  feeling  that  the  general  method  of  treatment  adopted  by 
Professor  Reye  best  serves  the  purposes  outlined  above.  On  the 
other  hand,  the  author  has  not  failed  to  consult  and  to  profit  by 
other  texts  on  Projective  Geometry  that  occupy  important  places 
in  recent  literature;  notably,  Veblen  and  Young,  Projective  Geome- 
try; Enriques,  Geometria  Proiettiva;  Severi,  Complementi  di  Geome- 
tria  Proeittiva. 

No  attempt  has  been  made  to  set  forth  a  necessary  and  sufficient 
set  of  postulates  for  Projective  Geometry;  not  that  the  author  fails 
to  recognize  the  importance  of  research  already  completed  in  this 
field,  but  because  of  the  conviction  that  the  student  is  unfitted  to 
appreciate  work  of  this  character  until  he  has  assimilated  the  main 
body  of  theorems  and  their  applications  based  upon  concepts 
familiar  to  him  from  the  study  of  Elementary  Geometry.  This, 


vi  PREFACE 

too,  is  in  accord  with  the  aims  set  forth  above.  The  existence  of 
ideal  elements  must  be  assumed  (Art.  7) ;  and  the  Dedekind  postu- 
late, or  an  equivalent,  must  be  used  in  order  to  arrive  at  continu- 
ously protective  forms.  The  treatment  of  the  Dedekind  postulate 
for  this  purpose  (Art.  39)  is  confessedly  meager,  and  many  teachers 
may  feel  the  need  of  expanding  it,  or  indeed  of  restating  it,  as 
occasion  seems  to  demand. 

No  attempt  has  been  made  to  introduce  new  or  strange  terms, 
the  only  exception,  so  far  as  the  author  is  aware,  is  the  use  of  the 
word  "confocal"  to  indicate  those  elements  of  a  double  polarity 
which  are  the  supports  of  coinciding  involutions  of  conjugate  ele- 
ments (Art.  159). 

While  this  book  has  grown  out  of  lectures  given  to  students  pre- 
paring to  teach  geometry,  the  subject  matter  is  by  no  means  of 
interest  to  this  class  of  students  alone.  The  engineer  and  the 
artisan  must  of  necessity  become  familiar  with  the  elementary 
processes  of  projection  and  section,  and  these  processes  are  the 
same  whether  they  lead  to  properties  of  geometrical  figures  or  to 
methods  in  mechanical  drawing. 

The  author  takes  this  occasion  to  express  his  gratitude  to  Pro- 
fessor Thomas  F.  Holgate,  now  Acting  President  of  Northwestern 
University,  for  inspiration  and  enthusiasm  acquired  under  his  in- 
struction at  Clark  University;  and  also  his  indebtedness  for  many 
helpful  suggestions  during  the  preparation  of  the  manuscript  for 
this  book. 

Especial  acknowledgment  is  due  to  Professor  Henry  S.  White, 
of  Vassar  College,  who  read  the  manuscript  and  whose  kindly 
comments  and  criticisms  have  materially  improved  the  book  in  a 
number  of  points. 

The  author  wishes  also  to  express  his  thanks  to  Professor  Charles 
S.  Slichter,  of  the  University  of  Wisconsin  for  his  sympathetic  in- 
terest during  the  preparation  of  the  manuscript  and  for  his  aid  in 
seeing  the  book  through  the  press. 

L.  WAYLAND  BOWLING. 
UNIVERSITY  OF  WISCONSIN, 
June,  1917. 


CONTENTS 

PAGE 

PREFACE v 

CHAPTER  I 

THE  ELEMENTS  AND  THE  PRIMITIVE  FORMS 

1.  Central  Projection 1 

2.  The  Elements .1 

3.  Notation 2 

4.  Primitive  Forms 2 

5.  Projection  and  Section 3 

6.  Projection  and  Section  Applied  to  the  Primitive  Forms  .    .  4 

7.  Ideal  Elements.    ..'..'. 5 

CHAPTER  II 
THE  PRINCIPLE  OF  DUALITY — SIMPLE  AND  COMPLETE  FIGURES 

8.  Duality  in  the  Plane 8 

9.  Duality  in  Space 9 

10.  The  Principle  of  Duality 10 

11.  Simple  Figures  in  a  Plane 10 

12.  Complete  Figures  in  a  Plane 12 

13.  Simple  and  Complete  Figures  in  Space .  13 

CHAPTER  III 

CORRELATION  OF  GEOMETRIC  FIGURES;  PERSPECTIVE  POSI- 
TION OF  GEOMETRIC  FIGURES 

14.  Correlation  of  Geometric  Figures 16 

15.  Perspective  Position  of  Geometric  Figures 16 

16.  Notation 19 

17.  Desargues  Theorem — Theorem  I 20 

18.  Converse  of  Theorem  I 22 

19.  Definitions 22 

vii 


iriii  CONTENTS 

PARE 

20.  Complete  Quadrangles  in  Perspective  Position — Theorem  II     23 
CHAPTER  IV 


21.  Harmonic  Ranges 25 

22.  Effect  of  Order 26 

23.  Construction  of  Harmonic  Ranges — Theorem  III 27 

24.  Harmonic  Pencils 28 

25.  Sections  of  Harmonic  Pencils — Theorem  IV 28 

26.  Harmonic  Conjugate  of  an  Ideal  Point 30 

27.  Normal  Conjugate  Rays  in  a  Harmonic  Pencil 30 

28.  Cross-ratio  of  a  Harmonic  Range 31 

29.  Harmonic  Mean  Between  Two  Numbers 32 

30.  Geometric  Mean  Between  Two  Numbers 33 

31.  The  Circle  of  Apollonius 34 

32.  Orthogonal  Circles 34 

CHAPTER  V 
PROJECTTVELY  RELATED  PRIMITIVE  FORMS  OF  THE  FIRST  KIND 

33.  Primitive  Forms  of  the  First  Kind 37 

34.  Chains  of  Perspectivity 37 

35.  Definition  of  Projective  Relationship 38 

36.  Harmonic  Scales 38 

37.  Two  Pairs  of  Points  Each  Harmonically  Separated  by  a 
Third  Pair— Theorem  V.    . 39 

38.  Converse  of  Theorem  V 40 

39.  Consequences  of  Theorem  V  and  Its  Converse — Theorem  VI.  41 

40.  Superposition — Self-corresponding  Elements  .......  44 

41.  Von  Staudt's  Fundamental  Theorem — Theorem  VII    ...  46 

42.  Consequences  of  Theorem  VII 47 

43.  Determination  of  Projective  Relationship — Theorem  VIII.  50 

CHAPTER  VI 
ELEMENTARY  FORMS 

44.  Definition  of  Elementary  Forms 55 


CONTENTS  ix 

PAGE 

45.  Fundamental  Properties  of  the  Curve  and  the  Envelope.    .  57 

46.  Construction  of  Curves  and  Envelopes 62 

47.  Relations  Existing  among  the  Elementary  Forms 64 

48.  Totality  of  Elementary  Forms 66 

49.  Classification  of  Curves  of  Second  Order 66 

50.  The  Conic  Sections 66 

CHAPTER  VII 

THE  PASCAL  THEOREM  AND  THE  BRIANCHON  THEOREM 

51.  Six  Elements  of  a  Conic  or  of  an  Envelope — Theorems  IX.  68 

52.  Converse  Theorems 69 

53.  Application  of  Theorems  IX 71 

54.  Degenerate  Cases  of  Theorems  IX 72 

55.  The  Pentagon  Theorem  and  Its  Dual 73 

56.  Application  of  the  Pentagon  Theorem  and  Its  Dual ....  74 

57.  The  Quadrangle  Theorem  and  Its  Dual 76 

58.  Application  of  the  Quadrangle  Theorem  and  Its  Dual.    .    .  76 

59.  The  Principle  of  Continuity 77 

60.  Second  Proof  of  the  Quadrangle  Theorem  and  Its  Dual  .    .  77 

61.  Generation  of  Particular  Conies  and  Envelopes 78 

62.  Cones  and  Sheaves  of  Planes  of  Second  Class 79 

63.  Cylinders 79 

CHAPTER  VIII 

POLES  AND  POLAR  LINES  WITH  RESPECT  TO  A  CURVE  OF 
SECOND  ORDER 

64.  Poles  and  Polar  Lines — Theorems  X 81 

65.  Special  Positions  of  Pole  and  Polar  Line 83 

66.  Chords  of  Contact 83 

67.  Construction  of  Poles  and  Polar  Lines 84 

68.  Conjugate  Points  and  Conjugate  Lines  with  Respect  to  a 
Conic— Theorems  XI 86 

69.  Consequences  of  Theorems  XI 87 

70.  Polar  Figures  with  Respect  to  a  Fixed  Conic 89 

71.  Self-polar  Figures 91 

72.  Pole-rays  and  Polar  Planes  with  Respect  to  a  Cone.    ...  91 


CONTENTS 


73.  Diameters  and  Centers  of  Conies 94 

74.  Conjugate  Diameters 94 

75.  Application  of  the  Harmonic  Properties  of  Poles  and  Polar 
Lines 95 

76.  The  Axes  of  a  Conic 97 

77.  The  Vertices  of  a  Conic 99 

78.  Algebraic  Equations  of  the  Conies 100 

79.  Diametral  Planes  and  Axes  of  Cylinders 104 

CHAPTER  X 

RULED  SURFACES  OF  SECOND  ORDER 

80.  Ruled  Surfaces— Theorem  XII 105 

81.  Sections  of  a  Surface  of  Second  Order 106 

82.  Tangent  Lines  and  Tangent  Planes 107 

83.  Tangent  Cones 108 

84.  Polar  Planes 108 

85.  Circumscribing  Tetrahedrons 109 

86.  The  Class  of  a  Ruled  Surface  of  Second  Order 109 

87.  Classification  of  Ruled  Surface  of  Second  Order 109 

CHAPTER  XI 
PROJECTTVELY  RELATED  ELEMENTARY  FORMS 

88.  Four  Harmonic  Elements  of  an  Elementary  Form    .    .    .    .  112 

89.  Elementary  Forms  in  Perspective  Positions  with  Primitive 
Forms 113 

90.  Elementary  Forms  in  Perspective  Positions  with  Each  Other  114 

91.  Protectively  Related  Forms 114 

92.  Determination  of  Projective  Relationship 115 

93.  Superposition  of  Projectively  Related  Forms 116 

94.  Double  Elements  of  a  Projectivity 116 

95.  The  Axis  of  a  Projectivity  on  a  Conic 117 

96.  The  Center  of  a  Projectivity  on  a  Conic 118 

97.  Double  Elements  of  a  Projectivity  on  any  Form 119 


CONTENTS  xi 

PAGE 

98.  Application  of  the  Theorem  in  Art.  95 119 

99.  Classification  of  Projectivities  on  a  Form .    .  120 

100.  Cyclic  Projectivities x 121 

101.  Construction  of  Cyclic  Projectivities 121 

CHAPTER  XII 

THE  THEORY  OF  INVOLUTION — IMAGINARY  ELEMENTS 

102.  Definition  of  Involution 127 

103.  Fundamental  Theorems— Theorem  XIII 127 

104.  Hyperbolic  Involutions 128 

105.  Elliptic  Involutions 128 

106.  Parabolic  Involutions 129 

107.  Involutions  on  a  Straight  Line 130 

108.  Involutions  on  a  Sheaf  of  Rays  of  First  Class 132 

109.  Involutions  Determined  by  a  Complete  Quadrangle,  or  a 
Complete  Quadrilateral 133 

110.  Desargues  Theorem  and  Its  Dual 134 

111.  Involutions  Determined  by  a  Fixed  Conic 135 

112.  Imaginary  Points  on  a  Straight  Line 137 

113.  Imaginary  Lines  in  a  Plane 138 

114.  Imaginary  Planes 139 

115.  Construction  Problems 139 

116.  Imaginary  Elements  on  any  Form 141 

117.  Construction  Problems 141 

CHAPTER  XIII 
THE  Foci  AND  FOCAL  PROPERTIES  OF  CONICS 

118.  Definition  of  Focus 146 

119.  Construction  of  Foci 146 

120.  Directrices  and  Focal  Radii 149 

121.  Fundamental  Theorem— Theorem  XIV 150 

122.  Consequences  of  Theorem  XIV 150 

CHAPTER  XIV 
PROJECTIVELY  RELATED  PRIMITIVE  FORMS  OF  THE  SECOND  KIND 

123.  Primitive  Forms  of  the  Second  Kind  .  .  159 


xii  CONTENTS 

PAGE 

124.  Perspective  Position  of  Planes  and  Bundles 159 

125.  Orthogonally  Correlated  Bundles 160 

126.  Definition  of  Projective  Relationship 160 

127.  Definition  of  Collineation 161 

128.  Definition  of  Duality 161 

129.  Consequences  of  the  Foregoing  Definitions 161 

130.  Fundamental  Theorem— Theorem  XV 161 

131.  Determination  of  Projective  Relationship 163 

132.  Projectivities  in  a  Form  of  Second  Kind 165 

133.  The  Perspectivity 165 

134.  Construction  of  a  Perspectivity  in  a  Plane 166 

135.  The  Invariant  of  a  Perspectivity  in  a  Plane 167 

136.  The  Harmonic  Perspectivity  or  Involution  in  a  Plane.    .    .  168 

137.  Limiting  Lines  in  a  Collineation 168 

138.  The  Affinity 170 

139.  Fundamental  Property  of  an  Affinity 170 

140.  Corresponding  Conies  in  Affinately  Related  Planes  ....  172 

141.  The  Area  of  a  Parabolic  Segment 173 

142.  The  Theorem  of  Appolonius 173 

143.  The  Similitude 176 

144.  Properties  of  the  Similitude 176 

145.  Inverse  Points.     Radical  Axis 177 

146.  The  Congruence 178 

147.  Collineation  in  the  Plane.     Self-corresponding  Elements.  179 

CHAPTER  XV 
POLARITIES  IN  A  PLANE  AND  IN  A  BUNDLE 

148.  The  Polarity  in  a  Plane 183 

149.  Construction  of  a  Polarity  in  a  Plane 184 

150.  Self-conjugate  Points  in  a  Polarity 185 

151.  Classification  of  Polarities  in  a  Plane 187 

152.  The  Polarity  in  a  Bundle 189 

153.  The  Orthogonal  Polarity 190 

154.  Polarity  and  Anti-polarity  with  Respect  to  a  Circle.    .    .    .  190 

155.  The  Absolute  Polarity 192 

156.  Double  Polarities  in  a  Plane  and  in  a  Bundle 192 

157.  Common  Elements  in  a  Double  Polarity 193 

158.  Double  Conjugate  Elements 194 

159.  Confocal  Elements  in  a  Double  Polarity 196 


CONTENTS  xiii 

PAGE 

160.  Construction  of  the  Confocal  Elements  of  a  Double  Polarity  196 

161.  Application  to  Cones  of  the  Second  Order 199 

162.  Cyclic  Planes  and  Focal  Axes  of  Cones 201 

163.  Quadric  Transformations 202 

164.  Perspective  Quadric  Transformations 203 

165.  Inversion  with  Respect  to  a  Circle 204 

166.  Properties  of  Positive  Inversions 205 

167.  Circular  Transformations 207 

168.  General  Note 208 

INDEX.  .  211 


PROJECTIVE  GEOMETRY 

CHAPTER  I 

THE  ELEMENTS  AND  THE  PRIMITIVE  FORMS 
PROJECTION  AND  SECTION 

1.  Central  Projection. — If  any  object,  for  example  a  flat  geo- 
metric figure,  is  projected  from  the  eye  and  a  plane  is  intercepted 
between  the  eye  and  the  object,  there  will  be  marked  out  on  this 
plane  a  new  figure  which  bears  certain  resemblances  to  the  original 
one.     The  figure  may  be  changed  in  size  and  shape,  but  the  rela- 
tive positions  of  lines  and  points  to  each  other  will  remain  un- 
altered.    If  two  lines  intersect  in  the  original  figure,  the  corre- 
sponding lines  will  intersect  in  the  new.     If  two  points  are  joined 
by  a  line  in  the  original  figure,  the  corresponding  points  will  lie 
on  the  corresponding  line  in  the  new.     The  process  of  obtaining 
the  new  figure  from  the  original  figure  is  called  central  projection. 

Projective  geometry  deals  primarily  with  those  properties  of 
figures  which  are  unaltered  by  central  projection.     Such  properties 
may  be  studied  by  analytical  methods  in  which  an  algebraic 
symbolism  is  used;  or,  as  in  this  book,  the  methods  of  pure  geome- 
try may  be  followed,  in  which  the  geometric  concept  is  kept 
constantly  in  mind.     Projective  pure  geometry,  because  of  its 
methods,  is  sometimes  called  Modern  Synthetic  Geometry,   a 
name  which  distinguishes  it  from  the  ancient  elementary  pure 
geometry  on  the  one  hand,  and  the  more  recent  analytical  geom- 
etry on  the  other.     Because  it  is  concerned  with  the  relative^ 
positions  of  the  parts  of  a  figure  and  not  at  all  with  their  magni-  I  . 
tudes,  the  projective  pure  geometry,  or  modern  synthetic  geometry,  f! 
is  known  also  as  the  Geometry  of  Position. 

2.  The  Elements. — The  point,  the  line,  and  the  plane  are  the 
elements  of  projective  geometry.     These  elements  are  undefined 
and  each  of  them  may  be  thought  of  as  existing  independently  of 

1 


2  PROJECTIVE  GEOMETRY  l§3 

the  others;  that  is,  the  line  may  be  thought  of  as  existing  without 
regard  to  the  points  lying  on  it,  and  the  plane  may  be  considered 
apart  from  the  points  and  lines  lying  on  it. 

j      The  line,  by  which  we  shall  always  mean  straight  line,  or  ray, 
|  and  the  plane  are  regarded  as  unlimited  in  extent. 

3.  Notation. — To  represent  the  elements  we  shall  adopt  as 
notation,  italic  capital  letters,  A,  B,  C, — to  denote  points;  small 
italics,  a,  6,  c, — to  denote  lines  or  rays;  and  small  Greek  letters, 
a,  /3,  7, — to  denote  planes.     The  line  through  two  points  A  and  B 
will  be  denoted  by  AB;  the  line  of  intersection  of  two  planes  a  and 

t    0  will  be  denoted  by  «/3;  the  plane  determined  by  the  point  A 
and  the  line  b  will  be  denoted  by  Ab;  the  point  determined  by  the 
Vjine  a  and  the  plane  /3  will  be  denoted  by  a/3  and  so  on. 

4.  Primitive  Forms. — Each  element  may  be  regarded  as  the 
base  or  support  of  an  indefinite  number  of  elements  of  another 
kind.    For  example,  a  plane  may  be  regarded  as  the  support  of 
the  indefinite  number  of  points  and  lines  which  lie  on  it;  a  line 
may  be  thought  of  as  the  support  of  the  indefinite  number  of 
planes  which  can  be  drawn  through  it,  or  of  the  indefinite  number 
of  points  which  can  be  placed  on  it.     Again,  through  a  point 

^  may  be  drawn  an  indefinite  number  of  planes  or  lines.     Such 
combinations  of  elements  are  known  as  primitive  forms.     They 
-1     ~ .'  -are  defined  as  follows : 

1.  The  point-row  or  range  of  points  is  the  aggregate  of  points 
-—lying  on  a  straight  line.     The  straight  line  is  the  base  or  support 

of  the  point-row  and  is  unlimited  in  extent. 

2.  The  sheaf  of  lines,  or  rays,  is  the  aggregate  of  lines  or  rays 
lying  in  one  plane  and  passing  through  one  point.     The  point  is 
the  center  or  support  of  the  sheaf. 

3.  The  sheaf  of  planes  is  the  aggregate  of  planes  passing  through 
one  line.     The  line  is  the  axis  of  the  sheaf. 

4.  The  field  of  points  is  the  totality  of  points  lying  in  one  plane. 
The  plane  is  the  support  of  the  field. 

5.  The  field  of  lines,  or  rays,  is  the  totality  of  lines,  or  rays, 
lying  in  one  plane.     The  plane  is  again  the  support  of  the  field. 

6.  The  bundle  of  lines,  or  rays,  is  the  totality  of  lines  or  rays 
passing  through  one  point.     The  point  is  the  center  or  support  of 
the  bundle. 


§5]  ELEMENTS  AND  PRIMITIVE  FORMS  3 

7.  The  bundle  of  planes  is  the  aggregate  of  planes  passing  through 
one  point.     The  point  is  again  the  center  or  support  of  the  bundle. 

8.  All  points  in  space  constitute  a  primitive  form  called  the 
space  of  points;  all  planes  in  space  form  the  space  of  planes;    Cf. 
all  the  lines  in  space  meeting  a  given  line  form  a  special  linear  Jo , 
complex  of  rays ;  and  all  lines,  or  rays,  in  space  form  the  space  of  /  / 1 
lines,  or  rays. 

There  are,  then,  in  all  eleven  distinct  primitive  forms  with 
which  we  have  to  deal. 

5.  Projection  and  Section. — Projection  and  section  are  the~7T 
common  processes  of  projective  geometry.     When  we  look  at  any 
object,  a  building  for  example,  every  visible  point  determines 
with  the  eye  a  ray  which  is  called  the  projector  of  the  poinfc^" 
Every  visible  straight  line  of  the  building  determines  with  the 
eye  a  plane  which  is  the  projector  of  the  line.     The  bundle  of  "^ 
which  the  eye  is  the  center  and  which  is  composed  of  the  projectors 
of  all  visible  points  and  lines  of  the  building  is  called  the  projector  3fc 
of  the  building.     The  projector  of  a  curved  line  on  the  building 
is  a  conical  surface  composed  of  the  projectors  of  all  the  points  of 
the  curve.     This  conical  surface  is  part  of  the  bundle  which  is  the  .' 
projector  of  the  entire  building. 

If  we  interpose  a  plane  between  the  eye  and  the  building,  and 
imagine  that  each  projector  makes  its  passage  through  the  plane 
visible,  we  shall  have  a  detailed  picture  of  the  building  on  the 
plane.     This  picture  is  a  section  of  the  bundle.     The  section  of 
an  individual  projector  by  the  plane  is  called  the  trace  of  that  ^ 
projector  on  the  plane  and  the  section  of  the  bundle  is  made  up 
of  the  traces  of  all  the  projectors  on  the  plane.     The  picture  or 
diagram  so  obtained  is  called  a  projection  of  the  building  on  the-^ 
plane.     In  general,  the  projector  of  a  point  from  a  point  is  a  ray, 
and  of  a  ray  is  a  plane.     The  projection  of  a  point  on  a  plane  is  a  -^ 
point,  and  of  a  line  is  a  line. 

The  process  of  projection  and  section  is  familiar  in  photography. 
Here  the  projectors  of  the  various  points  and  lines  of  the  object 
photographed,  drawn  to  the  lens  of  the  camera,  are  produced  back 
of  the  lens  and  leave  their  traces  on  the  sensitized  plate.  The 
photograph  obtained  is  a  section  of  the  bundle  of  projectors  and  a 
projection  of  the  object  photographed.  Again,  the  retinal  image 


PROJECTIVE  GEOMETRY 


[§6 


in  the  eye  is  a  section  of  the  bundle  of  projectors  from  the  land- 
scape at  which  we  are  looking,  the  center  of  the  bundle  being  the 
eye.  The  retinal  image  is  a  projection  of  the  landscape. 

The  draftsman  makes  use  of  central  projection  and  section 

\  when  he  constructs  the  elevation  of  a  dwelling.     Such  an  elevation 

',  T  •/  is  but  a  section  of  the  rays  drawn  from  an  assumed  position  of  the 

1  observer  to  all  the  visible  points  of  the  dwelling;  that  is,  it  is  a 

V  projection  of  the  dwelling. 

6.  Projection  and  Section  Applied  to  the  Primitive  Forms.— 
Let  u  be  any  point-row  containing  the  points  A,B,C,D,  .  .  .  , 
and  let  S  be  any  point  not  on  this  point-row.  Connect  S  with 
each  of  the  points  A,  B,  C,  D,  .  .  .  .  These  lines  are  the  pro- 


FIG.  1. 


jectors  of  the  points  of  the  point-row  from  the  center  S,  and 
when  each  is  produced  indefinitely  in  both  directions  we  obtain 
a  sheaf  of  rays  whose  center  is  S.  Hence  the  projector  of  a 
point-row,  from  a  point  not  lying  on  it,  is  a  sheaf  of  rays. 

Similarly,  if  we  project  a  sheaf  of  rays  whose  center  is  S,  from 
a  point  not  lying  in  its  plane,  from  the  eye  for  example,  we  ob- 
tain a  sheaf  of  planes  whose  axis  is  the  projector  of  the  center  S. 

If  we  project  a  field  of  points,  or  a  field  of  lines,  from  a  point 
not  lying  in  the  field,  we  obtain  a  bundle  of  rays,  or  a  bundle  of 
planes. 

Again,  if  we  cut  a  sheaf  of  rays  by  a  line  not  passing  through 


§7]  ELEMENTS  AND  PRIMITIVE  FORMS  5 

its  center  we  obtain  a  point-row,  and  if  we  cut  a  sheaf  of  planes 
l>y  a  plane  not.  passing  through  its  axis  we  obtain  a  sheaf  of  rays. 

In  the  same  way,  the  section  of  a  bundle  of  rays  is  a  field  of 
points  and  the  section  of  a  bundle  of  planes  is  a  field  of  lines. 
The  primitive  forms  can  thus  be  derived,  one  from  another,  by 
means  of  projection  and  section. 

Any  plane  figure  consisting  of  points  and  lines  can  be  projected 
from  a  point  S  not  lying  in  the  plane.  The  projector  thus  obtained 
consists  of  rays  and  planes  passing  through  S.  This  can  be  cut 
by  any  plane  not  passing  through  S,  and  a  second  plane  figure  is 
obtained.  We  say  that,  by  this  process,  the  first  figure  has  been 
projected  into  the  second,  or  that  the  second  figure  is  a  projection 
of  the  first. 

7.  Ideal  Elements. — In  Fig.  1,  suppose  that  a  ray,  unlimited 
in  extent,  rotates  about  S  in  counterclockwise  direction,  passing 
in  turn  through  the  positions  a,b,c,d,.  .  .  .  Its  intersection 
with  the  line  u  will  move  along  u  always  in  the  same  direction, 
passing  in  turn  through  the  positions  A,  B,  C,  D,  .  .  .  .  As  the 
rotation  continues,  the  point  of  intersection  of  this  ray  with  u 
is,  for  the  instant,  lost  to  view  far  to  the  right  and  immediately 
thereafter  it  appears  far  to  the  left  still  moving  along  u  in  the 
same  direction  as  before.  It  is  assumed  that  for  one  and  only 
one  position  of  the  rotating  ray,  the  intersection  with  u  is  lost  to 
the  senses,  or  is  not  in  the  finite  region.  In  ot]i££_words,  it  is  >/ 
assumed  that  on  the  straight  line  u  there  is  one  and  only  one 
infinitely  distant  point,  and  that  this  point  makes  the  line  con- 
tinuous from  extreme  right  to  extreme  left,  or  vice  versa.  This 
point  is  called  the  ideal  point  on  the  line,  other  points  on  the  line 
being  called  actual  points.  The  ideal  point  is  commonly  called 
the  infinitely,  distant  point  on  the  line. 

That  ray  of  the  sheaf  S  which  passes  through  the  ideal  point 
u  is  said  to  be  parallel  to  u.     Through  S  there  is  thus  one  and 
only  one  line  parallel  to  u. 

What  has  been  said  about  the  line  u  obviously  holds  for  any 
other  straight  line,  hence  we  have  the  fundamental  assumption  :\ 
On  every  straight  line  there  is  one  and  only  one  ideal  or  infinitely 
distant  point.     This  point  makes  the  line  continuous  from  any  one 
point  on  it  to  any  other  point  on  it  in  either  direction.     Through  a 


K 


6  PROJECTIVE  GEOMETRY  [§7 

given  point  there  can  be  drawn  one  and  only  one  line  parallel  to  a 
given  line.  This  parallel  intersects  the  given  line  in  the  ideal  or 
infinitely  distant  point. 

Suppose  that  in  a  plane  a  line  a  rotates  about  a  fixed  point  S. 
Every  point  of  a,  including  the  ideal  point,  traces  out  a  path.  The 
rotating  line  will  eventually  be  parallel  to  any  given  line  in  the 
plane  and,  in  this  position,  its  ideal  point  will  coincide  with  the 
ideal  point  on  the  given  line.  —  -Tiie4deaLpoiat  on  a  will  therefore 
eventually  coincide  with  each  ideal  point  in  the  plane  and  hence 
the  path  of  the  ideal  point  on  a  includes  all  the  ideal  points  in  the 
£lane.  This  locus  has  the  property  of  being  met  by  any  line  of 
the  plane  in  but  one  point,  and  hence  it  is  itself  a  straight  line. 
This  straight  line,  the  locus  of  all  the  ideal  points  in  the  plane,  is 
called  the  ideal  line,  or  the  infinitely  distant  line  of  the  plane. 
Two  parallel  planes  intersect  in,  or  have  in  common,  the  same 
ideal  line. 

Similarly,  the  locus  of  all  the  ideal  points  in  space  is  a  plane,  for 
if  it  were  a  curved  surface,  some  lines,  not  wholly  ideal,  would 
have  more  than  one  point  in  common  with  it;  that  is,  would 
have  more  than  one  ideal  point;  and  some  planes  would  cut  this 
surface  in  a  curved  line  and  thus  have  a  curved  locus  of  ideal 
points.  This  plane,  the  locus  of  all  the  ideal  points  in  space,  is  the 
ideal  plane  of  space. 

A  series  of  parallel  lines  in  a  plane  is  a  sheaf  of  rays  whose 
center  is  an  ideal  point.  A,  series  of  parallel  lines  in  space  is  a 
bundle  of  rays  whose  center  is  an  infinitely  distant  point.  A 
series  of  parallel  planes  is  a  sheaf  of  planes  whose  axis  is  an  ideal 

1,'y.Q^ 

The  introduction  into  geometry  of  the  ideal  elements  makes  it 
possible  to  include  in  a  general  statement  many  conditions  which 
otherwise  would  appear  as  exceptional  cases.  For  example,  any 
two  straight  lines  in  a  plane  intersect  in  a  point.  The  point  may 
be  actual  or  ideal.  Except  for  the  recognition  of  ideal  points, 
this  statement  would  read:  Any  two  straight  lines  in  a  plane 
intersect  in  a  point  unless  they  are  parallel.  Similarly,  any  two 
planes  intersect  in  a  straight  line  which  may  be  either  actual  or 
ideal.  If  the  line  of  intersection  is  ideal,  it  contains  all  the  ideal 
points  of  either  plane,  and  the  planes  are  parallel. 


§7]  ELEMENTS  AND  PRIMITIVE  FORMS  7 

Exercises 

1.  Project  any  two  lines  in  a  plane  into  two  lines  which  are  parallel 
to  each  other. 

2.  Project  a  sheaf  of  lines  into  a  system  of  parallel  lines. 

3.  Project  an  arbitrary  quadrangle  A  BCD  into  a  parallelogram. 
Suggestions. — Let  AD,  BC  meet  in  E  and  AB,  DC  in  F.    Let  S  be 

the  point  from  which  the  projection  is  made.     Cut  the  quadrangular 
pyramid  S-ABCD  by  a  plane  parallel  to  the  plane  SEF. 

4.  Project  any  two  lines  of  a  plane  into  two  perpendicular  lines. 
Suggestions. — Let  a  and  6  be  the  lines  and  A  and  B  be  two  points 

on  a  and  b  respectively.     Pass  a  second  plane  through  AB  and  in  it 
construct  a  right  triangle  on  AB  as  hypothenuse.     Let  S  be  the  right 
angle.     Project  from  S  and  cut  by  a  plane  parallel  to  SAB. 
6.  Project  any  plane  quadrangle  into  a  rectangle. 

6.  Project  any  plane  quadrangle  into  a  square. 

Suggestion. — A   rectangle   whose   diagonals   are   perpendicular   to 
each  other  is  a  square. 

7.  Project  any  triangle  into  an  equilateral  triangle. 
Suggestion. — Pass  a  second  plane  through  one  of  the  sides  and  in  it 

construct  an  equilateral  triangle  on  that  side  as  base.     Project  from 
the  vertex. 

8.  What  plane  figure  results  from  cutting  a  tetrahedron  by  an 
arbitrary  plane? 

9.  How  can  one  cut  a  cube  so  as  to  obtain  a  regular  hexagon? 


CHAPTER  II 

THE  PRINCIPLE  OF  DUALITY—  SIMPLE  AND  COMPLETE 

FIGURES 

8.  Duality  in  the  Plane.  —  We  have  already  noticed  (Art.  7) 
that  the  recognition  of  the  ideal  points  in  a  plane  makes  it  possible 
to  say,  without  exception: 

1.  Two  lines  a  and  b  in  a  plane  intersect,  and  so  determine,  the 
point  ab. 

Also  in  this  plane  : 

2.  Two  points  A  and  B  are  joined  by,  and  so  determine,  the  line 
AB. 

Comparing  statement  (1)  with  statement  (2),  we  see  that  either 
may  be  derived  from  the  other  by  a  simple  interchange  of  point 
with  line.  The  two  statements  are  said  to  be  plane-duals,  or 
reciprocals,  of  each  other.  The  elements  point  and  line  are 
dual  elements,  or  reciprocal  elements,  in  the  plane. 

The  interchange  of  dual  elements  may  be  illustrated  by  writing 
one  statement  over  the  other,  thus  : 

determine  the 


In  the  same  way,  from  any  geometrical  figure  consisting  of 
points  and  lines  in  a  plane,  can  be  derived  the  dual,  or  reciprocal, 
figure  by  interchanging  point  with  line.  The  following  are 
examples. 


1-  A"  the  haint    »«*'*» 

2.  All  the  [^ines8]  in  a  plane  constitute  a  field  of 

^]  not  [p^K^  point]    •» 
a  triangle. 


§9]  THE  PRINCIPLE  OF  DUALITY  9 

I 

Similarly,  any  theorem  involving  only  the  relative  positions  of 
points  and  lines  in  a  plane  has  a  dual,  or  reciprocal,  theorem  in- 
volving the  relative  positions  of  the  dual  elements,  line  and  point. 

Further  examples  of  the  use  of  duality  in  the  plane  will  appear 
as  we  proceed. 

9.  Duality  in  Space. — In  space  the  following  statements  hold. 

1. 


the 


4.  Two  which  have  a  common       Pln     determine  a 

f  plane  "I 
[  point  J  ' 

A  study  of  these  statements  shows  that  the  point  and  the 
plane  are  dual  elements  in  space,  while  the  line  is  its  own  dual, 
or  is  self-dual. 

Again, 

5.  All  the  [  .  P°in*J  on  a  1™?    1  constitute  a  [    P°fint/r?w    1  • 

Lplanes  through  a  line  J  Lsheaf  of  planesj 


6.  Anthe-  constitute  a 

L       lines  lying  m  a  plane       J  [_  field 

of  linesT 
of  linesj  ' 

7    All  the  T       Points  lying  in  a  plane       "I  PnriS5tifntpfl  f  field 

16  [planes  passing  through  a  point  J  C(  te  a  [bundle 

of  points'] 
of  planesj  ' 

From  these  examples  it  will  be  seen  that  each  primitive  form 
has  a  space-dual,  or  reciprocal,  primitive  form.  The  space  of 
points  and  the  space  of  planes  are  reciprocal  forms.  The  space 
of  lines  is  self-dual.  Also  the  special  linear  complex  of  rays  is 
self-dual  in  space. 

The  following  is  an  example  of  a  theorem  and  its  reciprocal 
placed  in  parallel  columns. 


10 


PROJECTIVE  GEOMETRY 


(§10 


If  four  points  A,  B,  C,  D,  are 
so  situated  that  the  lines  AB 
and  CD  intersect,  then  all  the 
points  lie  in  one  plane  and  con- 
sequently the  lines  AC  and  BD, 
and  also  the  lines  AD  and  BC, 
intersect. 


If  four  planes  a,  /3,  7,  8,  are 
so  situated  that  the  lines  «/3  and 
76  intersect,  then  all  the  planes 
pass  through  one  point  and  con- 
sequently the  lines  07  and  05, 
and  also  the  lines  a8  and  £7, 
intersect. 


10.  The  Principle  of  Duality. — The  principle  of  duality  asserts 
that  a  dual,  or  reciprocal,  statement  can  be  derived  from  a  given 
statement  as  in  the  foregoing  examples.     It  may  be  formulated 
as  follows : 

The  principle  of  duality  asserts  that  from  any  statement  or  theorem 
concerning  the  relative  positions  of  the  elements  composing  a  geo- 
metrical configuration,  another  statement  or  theorem  can  be  obtained 
by  a  simple  interchange  of  the  elements  of  the  configuration  with  their 
reciprocals. 

If,  for  example,  the  original  theorem  asserts  that  three  lines  in 
a  plane  meet  in  a  point,  then- the  reciprocal,  or  dual,  theorem  in 
the  plane  asserts  that  three  points  lie  on  a  line. 

Again,  if  the  original  theorem  states  that  four  planes  pass 
through  a  point,  the  reciprocal  theorem  asserts  that  four  points 
lie  in  a  plane. 

Exercises 

1.  What  is  the  plane-dual  of  a  triangle?     The  space-dual? 

2.  Show  that  the  dual  of  a  tetrahedron  is  another  tetrahedron; 
that  the  hexahedron  and  octahedron  are  dual  figures;  that  the  do- 
decahedron and  icosahedron  are  dual  figures. 

3.  What  plane  figure  is  obtained  by  cutting  a  cube  with  a  plane  not 
passing  through  any  vertex?     By  projecting  a  tetrahedron  from  any 
point  not  in  a  face? 

4.  If  the  points  A,  E,  C  lie  on  a  line  a  and  the  points  D,  B,  F  on  a 
second  line  b,  then  the  lines,  AB,  ED;  AF,  DC;  and  EF,  BC  intersect 
in  three  points  K,  L,  and  M  which  lie  on  a  third  line  c.     Write  out 
carefully  the  reciprocal  theorem  in  the  plane  and  draw  the  correspond- 
ing figure.     Also  write  out  the  space-dual  of  the  given  theorem. 

11.  Simple  Figures  in  a  Plane. — Besides  the  point-row  and  the 
sheaf  of  rays,  the  polygons  of  elementary  geometry  form  the 


§11] 


THE  PRINCIPLE  OF  DUALITY 


11 


simplest  combinations  of  points  and  lines  in  a  plane.  A  simple 
polygon  consists  of  a  number  of  points  (vertices)  arranged  in  some 
definite  order  and  joined  two  and  two  'in  this  order  by  straight 
lines  (sides),  the  last  point  being  joined  to  the  first.  The  polygon 
may  be  convex  or  reentrant, 
depending  upon  the  arrange- 
ment of  the  vertices  chosen. 
For  example,  from  the  four 
points,  A,  B,  C,  D  (Fig.  2) 
can  be  formed  the  three  simple 
quadrangles  A  BCD,  ACDB, 
and  ADBC.  Two  of  these 
quadrangles  are  reentrant, 
namely,  ACDB,  and  ADBC. 

A  simple  polygon  has  as 
many  sides  as  vertices  and  its 
dual  in  the  plane  is  another 
simple  polygon  having  an  equal 
number  of  sides  and  vertices. 

A  simple  polygon  having  n  vertices,  and  consequently  n  sides, 
is  often  called  a  simple  n-point  or  a  simple  n-side.  A  simple 
w-point  has  2n  elements,  namely,  the  n  vertices  and  the  n  sides. 


FIG.  2. 


FIG.  3. 

Corresponding  to  any  element  there  is  always  an  opposite  element. 
For  example,  in  the  simple  5-point,  shown  at  (a),  Fig.  3,  the  side 
8  is  opposite  the  vertex  3,  the  side  10  is  opposite  the  vertex  5, 
and  so  on.  In  the  simple  6-point,  shown  at  (6),  the  vertex  3 


12 


PROJECTIVE  GEOMETRY 


[§12 


is  opposite  the  vertex  9,  the  side  4  is  opposite  the  side  10,  and  so 
on.  In  general,  if  the  elements  of  any  simple  n-point  are  numbered 
in  order,  as  in  Fig.  3,  and  if  k  ^  n,  then  the  elements  k  and  A;  +  n 
are  opposite  each  other.  If  k  >  n,  then  k  and  k  —  n  are  opposite 
elements. 


FIG.  4. 

12.  Complete  Figures  in  a  Plane. — If  n  points  in  a  plane  are. 
joined  two  and  two  in  all  possible  ways,  a  figure  is  obtained  called 


FIG.  5. 

a  complete  n-point.  For  example,  if  the  four  points  A,  B,  C, 
D  (Fig.  4)  are  joined  in  pairs  in  all  possible  ways,  we  obtain  the 
six  lines  AB,  AC,  AD,  BC,  BD,  and  CD.  The  figure  is  a  complete 
4-point,  or  a  complete  quadrangle. 


§13]  THE  PRINCIPLE  OF  DUALITY  13 

Again,  if  n  lines  in  a  plane  intersect  two  and  two  in  all  possible 
ways,  the  figure  obtained  is  called  a  complete  n-side.  Thus,  in 
Fig.  5,  the  four  lines  a,  b,  c,  d  intersect  in  pairs  and  so  determine 
the  six  points  ab,  ac,  ad,  be,  bd,  and  cd.  The  figure  is  a  complete 
4-side,  or  complete  quadrilateral.  The  complete  quadrangle  and 
the  complete  quadrilateral  are  dual  figures  in  the  plane. 

A  complete  5-point,  or  complete  pentagon,  consists  of  5  points 
and  the  10  lines  joining  the  points  two  and  two.     The  dual  figure 
is  a  complete  5-side,  or  complete  pentalateral,  and  consists  of  5 
lines  together  with  their  10  points  of  intersection,  two  and  two. 
In  general, 


A  complete  n-point  consists  of 
n  points  in  the  plane  together 

with  all  the  ~        lines  join- 


A  complete  n-side  consists  of 
n  lines  in  the  plane  together 

n(n  —  1) 
with  all    the ~ —      points 


ing  the  points,  two  and  two.  of    intersection    of   the   lines, 

two  and  two. 

13.  Simple  and  Complete  Figures  in  Space. — The  projector  of 
a  simple,  or  a  complete,  plane  figure  from  a  point  not  in  its  plane, 
is  a  simple,  or  complete,  figure  in  space.  Thus,  the  projector  of 
a  simple  n-point  is  a  simple  n-edge ;  that  is,  an  ordinary  pyramid 
of  elementary  geometry  having  n  edges  and  n  faces.  The  pro- 
jector of  a  complete  n-point  is  a  complete  n-edge  having  n  edges 

and ~ — :-  faces;  and  the  projector  of  a  complete  n-side  is  a 

complete  n-face  having  n  faces  and ~ — "~  edges.     A  simple 

or  a  complete  n-edge,  or  n-face,  thus  consists  of  a  combination  of 
lines  and  planes  passing  through  a  point,  and  is,  therefore,  con- 
tained within  a  bundle  of  lines  or  a  bundle  of  planes. 

The  ordinary  polyhedrons  of  elementary  geometry  are  simple 
combinations  of  points,  lines,  and  planes.  Thus  an  octahedron 
has  6  points  (vertices),  12  lines  (edges),  and  8  planes  (faces). 
An  easy  generalization  leads  to  complete  combinations  of  points, 
lines,  and  planes.  Thus  a  complete  5-point  in  space  (Fig.  6) 
consists  of  5  points  (vertices)  together  with  the  10  lines  (edges) 
joining  the  points  two  and  two  and  the  10  planes  (faces)  de- 


14 


PROJECTIVE  GEOMETRY 


[§13 


termined  by  taking  the  points  in  threes.  The  reciprocal  figure 
is  a  complete  5-plane  consisting  of  5  planes  (faces)  together  with 
the  10  lines  (edges)  determined  by  taking  the  planes  in  pairs  and 
the  10  points  (vertices)  determined  by  taking  the  planes  in  threes. 
In  general, 

A  complete  n-point  in  space 

n(n  -  1) 
has         — ~ —         edges 


and 


n(n  -  1)  (n  -  2) 
6 


faces. 


A 

has 
n(n 

complete  n-plane  in  space 

2 
-D(n 

cugco        <*nu. 

6 

vertices. 

Many  interesting  and  important  combinations  of  points,  lines, 
and  planes  arise  from  a  st\idy  of  the  simple  and  complete  figures 
defined  in  this  and  the  preceding  articles.  As  an  example,  through 
each  edge  of  a  complete  5-point  in  space  there  pass  three  faces  and 
in  each  face  there  lie  three  edges.  Thus,  through  the  edge  AD 
(Fig.  6)  there  pass  the  faces  BAD,  BAD,  and  CAD;  and  in  the 
face  ABC  lie  the  three  edges  AB,  AC,  and  BC.  Hence,  if  we  cut 
the  5-point  by  a  plane  not  passing  through  any  vertex,  we  obtain 


§13]  THE  PRINCIPLE  OF  DUALITY  15 

a  plane  figure  consisting  of  10  lines  and  10  points  which  are 
respectively  the  traces  of  the  10  faces  and  the  10  edges.  In  this 
plane  figure,  three  lines  pass  through  each  point  and  three  points 
lie  on  each  line.  This  combination  of  points  and  lines  in  a  plane, 
thus  shown  to  be  possible,  is  known  as  the  configuration  of 
Desargues.  It  is  of  great  importance  in  the  development  of 
projective  geometry  from  the  point  of  view  we  have  chosen. 

Exercises 

1.  Construct  a  triangle  having  one  vertex  at  infinity;  having  two 
vertices  at  infinity. 

2.  ABCDE  is  a  simple  pentagon,   D  and  E  being  ideal  points. 
Construct  the  figure  and  point  out  the  vertex  opposite  the  infinitely 
distant  side;  the  sides  opposite  the  ideal  vertices. 

3.  Construct  a  complete  5-point  in  the  plane,  two  vertices  being 
at  infinity. 

4.  Given  any  plane  figure,  show  that  the  space-dual  of  this  figure 
is  a  projector  of  its  plane-dual. 

.  6.  Construct  a  complete  6-point  in  space.  Describe  a  section  of 
this  6-point  by  a  plane  not  passing  through  any  vertex.  Describe 
the  reciprocal  configuration  in  the  plane. 

6.  Show  that  the  plane-dual  of  a  configuration  of  Desargues  is 
another  configuration  of  Desargues. 

7.  Construct  a  complete  5-plane  in  space.     What  is  a  section  of 
this  figure  by  a  plane  not  passing  through  any  vertex? 

NOTE. — The  principle  of  duality  is  due  to  two  French  mathe- 
maticians, Poncelet  (1822)  and  Gergonne  (1826).  Desargues 
(1593-1662)  was  one  of  the  founders  of  modern  geometry. 


CHAPTER  III 

CORRELATION  OF  GEOMETRIC  FIGURES 
PERSPECTIVE  POSITION  OF  GEOMETRIC  FIGURES 

14.  Correlation  of  Geometric  Figures. — Any  combination  of 
the  elements  is  a  geometric  figure.  Primitive  forms,  simple 
figures,  and  complete  figures  are  thus  geometric  figures.  The 
points,  lines,  and  planes  of  a  geometric  figure  are  its  elements. 
Two  elements  of  a  geometric  figure  that  lie  on,  or  pass  through, 
each  other  are  called  incident  elements.  For  example,  if  the 
point  A  lies  on  the  line  a,  the  elements  A  and  a  are  incident. 
Again,  if  the  plane  a  passes  through  the  line  a,  a  and  a  are  incident 
elements.  Each  point  of  a  geometric  figure  is  thus  incident  to 
all  the  lines  and  planes  of  the  figure  passing  through  it;  each  line 
is  incident  to  all  the  points  of  the  figure  lying  on  it  and  to  all  the 
planes  of  the  figure  passing  through  it;  and  each  plane  is  incident 
to  all  the  points  and  lines  of  the  figure  lying  in  it. 

If  two  geometric  figures  are  so  related  that  to  any  element  of 
either  corresponds  a  definite  element  of  the  other  and,  further, 
if  incident  elements  in  either  always  correspond  to  incident 
elements  in  the  other,  then  the  figures  are  said  to  be  correlated 
one  to  one. 

Corresponding  elements  in  correlated  figures  are  often  called 
homologous  elements. 

As  an  example;  if  in  two  correlated  quadrangles,  the  vertices 
A,B,C,D  of  one  correspond  respectively  to  the  vertices  A\,  B\, 
Ci,  Dl  of  the  other,  then  any  side,  as  AB,  of  the  one  corresponds 
to  the  side  AiBi  of  the  other,  since  AB  is  incident  to  both  A  and 
B  and  must  correspond,  by  definition,  to  the  side  A\B\,  incident 
to  both  Ai  and  BI.  Similarly,  AC,  Aid;  AD,  AiDi;  etc.,  are 
pairs  of  homologous  sides. 

15.  Perspective  Position  of  Geometric  Figures.— Two  primi- 
tive forms  are  said  to  be  unlike  if  one  is  formed  of  elements  differ- 
ing in  kind  from  the  elements  composing  the  other.  Thus  a 

16 


§15]          CORRELATION  OF  GEOMETRIC  FIGURES  17 

point-row  and  a  sheaf  of  lines  are  unlike  primitive  forms.  Like 
primitive  forms  are  each  composed  of  the  same  kind  of  elements. 
Two  bundles  of  rays  are  like  primitive  forms. 

Two  unlike  primitive  forms  are  in  perspective  position  with 
each  other  if  one  is  a  section  of  the  other.  Any  element  of  the 
one  form  then  lies  on,  or  passes  through,  its  homologous  element 
of  the  other.  Thus  a  field  of  points  and  a  bundle  of  rays  are  in 


FIG.  7. 

perspective  position  when  the  first  is  a  section  of  the  second,  each 
point  of  the  field  lying  on  its  homologous  ray  of  the  bundle. 

Two  like  primitive  forms  are  in  perspective  position  if  each  is 
a  section  of,  or  a  projector  of,  the  same  third  primitive  form. 
Corresponding  elements  have  in  common  an  element  of  the  third 
primitive  form.  For  example,  two  sheaves  of  rays  are  in  per- 
spective position  if  they  are  each  projectors  of  the  same  point- 
row,  and  then  corresponding  rays  have  in  common  a  point  of 
the  point-row  (Fig.  7).  Two  sheaves  of  rays  are  also  in  perspec- 
tive position  if  each  is  a  section  of  the  same  sheaf  of  planes,  and 
then  corresponding  rays  have  in  common  a  plane  of  the  sheaf  of 
2 


18 


PROJECTIVE  GEOMETRY 


planes.  Since  the  planes  in  which  lie  the  sheaves  of  rays  meet  in 
a  line,  the  sheaves  of  rays  are  projectors  of  the  point-row  cut  from 
this  line  by  the  sheaf  of  planes  (Fig.  8). 


FIG.  8. 

Two  point-rows  are 'in  perspective  position  if  each  is  a  section 
of  the  same  sheaf  of  rays,  corresponding  points  then  have  in 


FIG.  9. 

common  a  ray  of  the  sheaf  (Fig.  9).  Two  non-intersecting  point- 
rows  are  in  perspective  position  if  each  is  a  section  of  .the  same 
sheaf  of  planes,  corresponding  points  then  have  in  common  a 
plane  of  the  sheaf. 


§16]          CORRELATION  OF  GEOMETRIC  FIGURES  19 

Clearly  it  is  not  always  possible  for  two  primitive  forms  to  be 
in  perspective  position.  Thus  a  sheaf  of  rays  and  a  field  of  points 
cannot  be  in  perspective  position,  for  the  one  cannot  be  a  pro- 
jector of  the  other.  Again,  a  point-row  and  a  field  of  points 
cannot  be  sections  of  any  third  primitive  form.  When  two 
primitive  forms  can  be  in  perspective  position,  they  are  said  to 
have  the  same  dimensions.  Thus,  the  first  three  primitive 
forms  (Art.  4)  are  called  primitive  forms  of  one  dimension. 
Each  can  be  in  perspective  position  with  either  of  the  other 
two,  and  cannot  be  in  perspec- 
tive position  with  any  of  the 
other  primitive  forms.  The 
primitive  forms  4,  5,  6,  7  (Art. 
4)  are  two-dimensional  primi- 
tive forms;  the  space  of  points, 
the  space  of  planes,  and  the 
special  linear  complex  are  three 
dimensional  primitive  forms; 
and  the  space  of  rays  has  four 
dimensions. 

Two  correlated  polygons  are 
in  perspective  position  if  the 
lines  joining  corresponding 
vertices  meet  in  a  point.  For  example,  two  triangles  are  in 
perspective  position  if  each  is  a  section  of  the  same  3-edge. 
If  the  triangles  lie  in  the  same  plane,  they  are  in  perspective 
position  when  the  lines  joining  corresponding  vertices  meet  in 
a  point  (Fig.  10). 

16.  Notation. — The  symbol    ^    is  used  to  designate    "is  in 
perspective  position  with."     Thus: 

The  sheaf  of  rays  S  X  ttie  point-row  u, 

is  read,  "the  sheaf  of  rays  S  is  in  perspective  position  with  the 
point-row  u,"  and  means  that  u  is  a  section  of  S.     Again: 

The  triangle  ABC  A  the  triangle  AiBid, 

is  read,  "the  triangle  ABC  is  in  perspective  position  with  the 
triangle  A\BiC\"  and  means  that  the  lines  joining  corresponding 


20  PROJECTIVE  GEOMETRY  [§17 

vertices  meet  in  a  point.     When  no  ambiguity  can  occur,  these 
statements  may  be  abbreviated  thus, 

S  7  u,  and  ABC  A  AiBiCi. 
Exercises 

1.  If  u  "A  S,  what  is  the  reciprocal  figure  in  space? 

2.  ABC  and  AiBiCi  are  two  correlated  triangles  in  the  same  plane 
and  in  perspective  position.     Construct  the  figure.     What  is  the  dual 
figure  in  the  plane?     In  space? 

3.  If  a  and  /3  are  two  fields  of  points  and  a  X  /3,  what  can  be  said 
of  lines  joining  corresponding  points?     Describe  the  dual  figure. 

4.  Under  what  condition  will  two  correlated  fields  of  rays  be  in 
perspective  position? 

6.  If  S  and  >S'  are  two  correlated  sheaves  of  rays  and  5  ^  S', 
what  is  the  locus  of  the  intersection  of  corresponding  rays? 

6.  Two  non-intersecting  point-rows  are  in  perspective  position. 
Describe  the  dual  figure. 

17.  Theorem  I. — Desargues  Theorem. — //  two  correlated  tri- 
angles are  so  situated  that  corresponding  sides  meet  in  three  points 
of  a  straight  line,  then  the  triangles  are  in  perspective  position. 

Let  the  vertices  A,B,  C  of  one  triangle  correspond  respectively 
to  the  vertices  Ai,Bi,Ci  of  the  other;  and  corresponding  sides 
meet  in  the  points  D,  E,  F  of  the  line  u. 

If  the  triangles  lie  in  different  planes  (Fig.  11),  a  pair  of  corre- 
sponding sides,  as  AB  and  AiBi,  determines  a  plane.  The  three 
planes  so  determined  are  the  faces  of  a  triangular  pyramid  whose 
edges  are  the  lines  joining  corresponding  vertices.  The  triangles 
are  thus  sections  of  the  same  3-edge  and  consequently  in  per- 
spective position. 

If  the  triangles  lie  in  the  same  plane  (Fig.  12),  pass  a  second 
plane  through  the  line  u  and  in  it  construct  a  third  triangle  AzBzCz 
so  that  the  sides  B2C2,  AzC2,  A2B2  pass  through  D,E,F,  respec- 
tively. Then  ABC  X  AzBzCz,  the  lines  joining  coVresponding 
vertices  meeting  in  0;  and  AiBiCi  7t  AzBzC*,  the  lines  joining 
corresponding  vertices  meeting  in  0'.  The  three  planes  A...l,.l, 
B->BiB,  and  C^CiC  form  three  planes  of  the  sheaf  of  planes  whose 
axis  is  00',  and  intersect  the  plane  of  the  original  triangles  in  the 
lines  AAi,  BBi,  and  CCi,  respectively.  These  lines  must,  there- 


}17]          CORRELATION  OF  GEOMETRIC  FIGURES  21 

s 

X. 

D 


FIG    12. 


22  PROJECTIVE  GEOMETRY  [§18 

fore,  pass  through  the  point  where  the  axis  00'  meets  the  plane 
ABC.     Hence  ABC  A  AiBid. 

18.  Converse  of  Theorem  I. — //  two  correlated  triangles  are 
in  perspective  position,  then  corresponding  sides  meet  in  three  points 
of  a  straight  line. 

If  the  triangles  do  not  lie  in  the  same  plane,  they  are  sections  of 
the  same  3-edge  and  corresponding  sides  must  meet  in  points  of 
the  line  common  to  their  planes  (Fig.  11). 

If  the  triangles  lie  in  the  same  plane,  let  the  line  joining  corre- 
sponding vertices  meet  in  S  (Fig.  12).  Through  S  draw  a  line 
not  lying  in  the  plane  of  the  triangles,  and  upon  this  line  choose 
two  points  as  0  and  0'.  Project  ABC  from  0  and  AiBiCi  from 
0',  thus  obtaining  two  triangular  pyramids  whose  edges  intersect 
in  pairs.  For  example,  BO  and  BiO'  intersect,  since  the  four 
points  B,  0, 0',  BI  lie  in  the  plane  BSO.  Similarly,  the  other 
pairs  of  edges  meet  and  we  have  a  third  triangle  A2B2C2  which  is 
in  perspective  position  with  each  of  the  given  triangles.  The 
sides  of  A2B2C2  must,  therefore, 'meet  the  corresponding  sides  of 
each  of  the  others  in  points  of  the  same  straight  line.  This  line 
is  the  intersection  of  the  plane  A2B2C2  with  the  plane  in  which  lie 
the  original  triangles. 

19.  Definitions. — If  the  triangle  ABC  ^the  triangle  AiBiCi, 
lines  joining  corresponding  vertices  meet  in  the  center  of  perspec- 
tivity,  and  corresponding  sides  intersect  upon  the  axis  of  perspec- 
tivity.     Thus  S  is  the  center  of  perspectivity  and  u  is  the  axis  of 
perspectivity  in  Figs.  11  and  12. 

Points  lying  upon  the  same  straight  line  are  collinear.  Lines 
passing  through  the  same  point  are  concurrent.  Planes  through 
the  same  line  are  coaxial.  Figures  in  the  same  plane  are  coplanar. 

Two  coplanar  triangles  which  are  in  perspective  position  form 
a  configuration  of  Desargues  (Art.  13).  Thus  the  10  points  and 
10  lines  lying  in  the  plane  ABC  (Fig.  12)  form  the  section  of  the 
complete  5-point  OA2B2C20'  by  the  plane  ABC. 

Exercises 

1.  Join  the  middle  points  of  the  sides  of  any  triangle  ABC  forming 
a  second  triangle,  AiBiCi.  The  two  triangles  are  ki  perspective 
position.  Where  is  the  axis  of  perspectivity?  The  center  of  perspec- 
tivity? Does  this  prove  that  the  medians  meet  in  a  point? 


§20]          CORRELATION  OF  GEOMETRIC  FIGURES  23 

2.  To  join  a  given  point  to   the  inaccessible  intersection  of  two 
given  lines. 

SOLUTION. — Let  u  and  Ui  be  the  lines  and  A  the  point.  Through 
a  point  0,  not  lying  on  either  line,  draw  three  lines  meeting  the  given 
lines  in  BC,  RS,  and  Bid.  Connect  A  with  B  and  C  determining 
upon  RS  the  points  M  and  N  respectively.  Draw  the  lines  CiN 
and  BiM  meeting  in  A\.  The  two  triangles  ABC  and  AiBiCi  are  in 
perspective  position.  Complete  the  solution. 

3.  Given  two  lines  a  and  b.     Through  a  point  P,  not  lying  on  either, 
draw  a  series  of  lines  forming  with   a  and  b  a  set  of  quadrangles. 
Show  that  the  diagonals  of  these  quadrangles  all  intersect  upon  the 
same  straight  line  which  also  passes  through  the  intersection  of  a  and  b. 

Suggestion. — The  point  P  is  the  center  of  perspectivity  for  a  series 
of  triangles  whose  axes  of  perspectivity  all  coincide  in  the  line  sought. 
This  line  has  been  called  the  polar  line  of  the  point  P  with  respect  to 
the  two  lines  a  and  b  (Poncelet). 

4.  If  a  point  P  is  connected  to  the  vertices  of  a  given  triangle  ABC, 
the  joining  lines  meeting  the  sides  opposite  in  AiBiCi,  respectively, 
the   two   triangles   ABC   and   AiBiCi   are  in   perspective   position. 
Construct  the  axis  of  perspectivity.     This  axis  is  often  called  the 
polar  line  of  the  point  with  respect  to  the  triangle  ABC.     The  point 
P  is  called  the  pole.     Construct  the  pole  of  a  given  line. 

6.  Show  that  in  the  configuration  of  Desargues  any  line  may  be 
taken  as  an  axis  of  perspectivity.  The  two  triangles  and  the  center 
of  perspectivity  will  then  be  uniquely  determined.  Also  that  any 
point  may  be  taken  as  center  of  perspectivity.  The  two  triangles 
and  the  axis  of  perspectivity  will  then  be  uniquely  determined. 

6.  Project  the  figure  in  exercise  3  so  that  the  point  P  and  the 
intersection  of  a  and  b  project  into  ideal  points.  Prove  the  property 
stated  in  the  exercise  for  the  new  figure  by  elementary  geometry. 

20.  Complete  Quadrangles  in  Perspective  Position.— Theorem 

II. — //  two  correlated  complete  quadrangles  are  so  situated  that  five 
pairs  of  corresponding  sides  meet  in  points  of  the  same  straight  line, 
then  the  sixth  pair  also  meet  on  this  line  and  the  quadrangles  are 
in  perspective  position. 

Let  ABCD  and  AiB±C\Di  be  two  quadrangles  so  situated  that 
the  five  pairs  of  corresponding  sides,  aoi,  661,  cc\,  dd\,  and  ee\ 
intersect  upon  u  (Fig.  13).  Then  (theorem  I),  ABC  A"  A\BiC\ 
and  BCD  A"  BidDi,  the  centers  of  perspectivity  coinciding  in 
S.  The  quadrangles  are,  therefore,  in  perspective  position. 
Moreover,  ACD  ^  AiCiDi,  and  consequently,  by  virtue ^of  the 
converse  of  theorem  I,  AD  and  A\D\  meet  on  u. 


24 


PROJECTIVE  GEOMETRY 


[§20 


FIG.  13. 


Exercises 

1.  If  two  complete  quadrangles  have  five  pairs  of  corresponding 
sides  parallel,  each  to  each,  show  that  the  sixth  pair  will  be  parallel. 

2.  Given  in  a  plane  a  fixed  parallelogram.     By  use  of  the  ruler  alone, 
draw  a  parallel  to  any  given  line  through  a  given  point. 

Suggestions. — Let  u  be  any  given  line  and  A  a  given  point.  Let 
u  meet  the  sides  of  the  parallelogram  in  MM'NN'  and  the  diagonal 
Did  in  P.  Join  A  to  M  and  M '  and  through  P  draw  any  line  cutting 
these  lines  in  C  and  D  respectively.  Join  D  and  C  to  2V  and  N' 
respectively  and  let  these  lines  meet  in  B.  AB  is  the  parallel  sought. 
For  the  quadrangle  A  BCD  is  in  perspective  position  with  the  quad- 
rangle whose  vertices  are  the  two  ideal  points  on  the  sides  of  the 
parallelogram  together  with  the  points  C\D\  (Lambert,  1774). 

3.  Two  triangles  are  in  perspective  position  but  are  not  coplanar. 
What  is  the  dual  figure? 

4.  Two   complete   quadrangles   are   coplanar   and   in   perspective 
position.     What  is  the  dual  figure  in  the  plane? 

6.  Show  that  any  two  complete  n-points  will  be  in  perspective  posi- 
tion provided  that  the  n-2  triangles  that  can  be  formed,  having  two 
of  the  vertices  of  one  n-point  as  common  vertices,  are  in  perspective 
position  with  their  corresponding  triangles  in  the  other  n-point. 


CHAPTER  IV 
HARMONIC  RANGES  AND  HARMONIC  PENCILS 

21.  Harmonic  Ranges. — If  four  points  A,B,C,D  are  so  situated 
along  a  line  that  two  sides  of  a  simple  quadrangle  intersect  in  A, 
the  two  remaining  sides  intersect  in  C,  one  diagonal  passes  through 
B,  and  the  other  diagonal  passes  through  D,  then  the  four  points 
are  called  four  harmonic  points  and  together  form  a  harmonic 
range.  The  range  is  said  to  be  defined  by  the  quadrangle.  Thus 
(Fig.  14),  ABCD  is  a  harmonic  range  defined  by  the  quadrangle 


KLMN,  since  the  sides  MN,  KL  meet  in  A,  the  sides  ML,  NK 
meet  in  C,  the  diagonal  NL  passes  through  B,  and  the  diagonal 
MK  passes  through  DJ  Also,  MOKD  is  a  harmonic  range  de- 
fined by  the  quadrangle  NALC,  since  NA,  LC  meet  in  M ;  AL,  NC 
meet  in  K;  the  diagonal  NL  passes  through  0;  and  the  diagonal 
AC  passes  through  D.  Similarly,  NOLB  is  a  harmonic  range 
defined  by  the  quadrangle  KAMC. 

The  points  of  a  harmonic  range,  taken  in  order  from  left  to 
right,  are  called  first,  second,  third,  and  fourth  harmonic  points. 
The  quadrangle  which  defines  the  range  has  two  sides  meeting 

25 


26  PROJECTIVE  GEOMETRY  [§22 

in  the  first  harmonic  point,  two  meeting  in  the  third,  one  diagonal 
passing  through  the  second,  and  one  diagonal  through  the  fourth. 
22.  Effect  of  Order. — It  is  clear,  from  the  definitions  in  the 
preceding  article,  that  if  ABCD  is  a  harmonic  range,  so  is  CBAD 
and  also  ADCB.  In  fact,  we  may  interchange  the  first  with  the 
third,  or  the  second  with  the  fourth,  or  make  both  these  inter- 
changes at  the  same  time  and  the  resulting  range  will  be  har- 
monic and  will  be  defined  by  the  same  quadrangle  KLMN  (Fig. 
15).  In  this  way  KLMN  defines  four  harmonic  ranges;  namely, 
ABCD,  ADCB,  CBAD,  and  CDAB.  But  this  is  not  all,  for  if 
the  diagonals  of  the  quadrangle  intersect  in  0,  then  the  triangles 


OBD  and  CMN  are  in  perspective  position  since  corresponding 
sides  meet  on  AK  (theorem  I).  Hence,  the  lines  OC,  BM,  and 
DN  meet  in  a  point  S  thus  forming  a  quadrangle  OMSN.  This 
quadrangle  defines  the  harmonic  ranges  DCBA,  DABC,  BCD  A, 
and  BADC.  Hence  we  can  say:  Of  the  24  possible  arrangements 
of  four  points  on  a  line,  8  are  harmonic  ranges  if  one  is  a  harmonic 
range. 

Of  these  8  harmonic  ranges  it  is  to  be  noted  that  A  and  C  are 
always  separated  by  B  and  D;  that  is,  it  is  not  possible  to  pass 
continuously  from  A  to  C  in  either  direction  without  passing  over 
either  B  or  D.  In  none  of  the  remaining  16  possible  arrangements 
are  A  and  C  separated  by  B  and  D  as  the  reader  can  readily  verify. 


§23] 


HARMONIC  RANGES  AND  PENCILS 


27 


In  any  harmonic  range  ABCD,  A  and  C,  and  also  B  and  D,  are 
called  harmonic  conjugates.  One  pair  of  harmonic  conjugates 
always  separates  the  other  pair.  Either  pair  is  said  to  be 
harmonically  separated  by  the  other. 

23.  Construction  of  Harmonic  Ranges. — Theorem  III. — Three 
points  of  a  straight  line  may  be  chosen  arbitrarily  as  the  first,  second, 
and  third  of  a  harmonic  range,  the  fourth  harmonic  point  is  then 
uniquely  determined. 


M 


FIG.  16. 

For  if  A,  B,  C  are  the  points  chosen  arbitrarily,  we  can  draw 
any  two  lines  through  A  and  any  line  through  B  meeting  the 
lines  through  A  in  L  and  N  (Fig.  16).  By  joining  L  and  N  to 
C  a  quadrangle  KLMN  is  formed  whose  diagonal  MK  determines 
the  fourth  harmonic  point  D.  Any  other  quadrangle  as  KiLiMiNi, 
constructed  like  KLMN,  will  determine  the  same  point  D,  for 


since  five  pairs  of  corresponding  sides  of  the  complete  quadrangles 
KLMN  and  K  \L\M\N\  meet  on  the  line  ABC  by  construction. 


28 


PROJECTIVE  GEOMETRY 


The  sixth  pair,  namely,  MK  a,ndM\Ki,  must,  therefore,  meet  on 
this  line  (theorem  II). 

24.  Harmonic  Pencils. — The  projector  of  a  harmonic  range, 
from  any  point  not  on  the  line  containing  the  range,  is  called  a 
harmonic  pencil  of  rays.  The  point  common  to  the  rays  is  the 
vertex  of  the  pencil. 


FIG.  17. 

The  projector  of  a  harmonic  pencil  of  rays,  from  any  point  not 
on  the  plane  of  the  pencil,  is  a  harmonic  pencil  of  planes.  The 
line  common  to  the  planes  is  the  axis  of  the  pencil. 

25.  Sections  of  Harmonic  Pencils. — Theorem  IV. — The  sec- 
tion of  a  harmonic  pencil  of  rays  by  any  line  not  passing  through 
its  vertex  is  a  harmonic  range  of  points.  The  section  of  a  harmonic 
pencil  of  planes  by  any  plane  not  containing  its  axis  is  a  harmonic 
pencil  of  rays. 


§25]  HARMONIC  RANGES  AND  PENCILS  29 

Let  ABCD  be  a  harmonic  range  denned  by  the  quadrangle 
KLMN  (Fig.  17).  Project  the  entire  figure  from  a  point  S  not 
on  the  plane  KLMN,  thus  obtaining  a  quadrangular  pyramid 
S-KLMN  and  a  harmonic  pencil  of  rays  S-ABCD.  Opposite 
faces  of  the  pyramid  intersect  in  SA  and  also  in  SC;  the  diagonal 
planes  pass  through  SB  and  SD.  Hence  a  section  of  this  figure, 
by  any  plane  not  passing  through  S,  consists  of  a  quadrangle 
KiLiMiNi  which  defines  the  harmonic  range  AiB\C\Di.  But 
this  range  is  any  section  of  the  harmonic  pencil  S-ABCD. 

The  pencil  of  planes  SN-ABCD  is  by  definition  a  harmonic 
pencil  of  planes  since  it  is  the  projector  of  the  harmonic  pencil  of 
rays  N-ABCD  from  the  point  S.  Any  section  of  this  harmonic 
pencil  of  planes  is  a  harmonic  pencil  of  rays,  since  the  cutting 
plane  will  meet  the  pencil  N-ABCD  in  a  harmonic  range,  and 
the  section  of  the  pencil  of  planes  is  a  projector  of  this  range. 

In  the  harmonic  pencil  of  rays  S-ABCD,  the  rays  SA  and  SC 
are  harmonically  separated  by  the  rays  SB  and  SD  and  vice 
versa.  Similarly,  in  the  pencil  of  planes  SN-ABCD,  the  planes 
SNA  and  SNC  are  harmonically  separated  by  the  planes  SNB 
and  SND. 

Exercises 

1.  Given  any  three  rays  of  a  sheaf  of  rays,  construct  the  fourth 
harmonic  ray. 

Suggestion. — Cut  the  three  given  rays  by  any  line  in  the  points  A, 
B,  C.  Use  theorem  III. 

2.  Given  three  planes  of  a  sheaf  of  planes,  construct  the  fourth 
harmonic  plane. 

3.  A  line  rotates  about  a  fixed  point  A  meeting  two  fixed  coplanar 
lines,  b  and  c,  in  the  variable  points  B  and  C  respectively.     What  is 
the  locus  of  the  fourth  harmonic  point  to  A,  B,  C?     Consider  the  case 
when  b  and  c  are  parallel. 

4.  A  line  passes  through  a  fixed  point  A  and  meets  two  fixed  planes 
in  the  variable  points  B  and  C:     What  is  the  locus  of  the  fourth 
harmonic  point? 

5.  Given  three  fixed  lines  a,  b,  c,  no  two  of  which  intersect.     A 
fourth  line  moves  so  as  to  meet  the  fixed  lines  in  the  variable  points 
A,  B,   C  respectively.     What  is  the  locus  of  the  fourth  harmonic 
point? 


30 


PROJECTIVE  GEOMETRY 


METRIC   PROPERTIES   OF   HARMONIC   RANGES   AND    HARMONIC 

PENCILS 

26.  Harmonic  Conjugate  of  an  Ideal  Point. — Let  D  be  the  ideal 
point  on  the  point-row  AC  (Fig.  18).  Through  A  draw  any  two 
lines  meeting  the  ideal  line  in  the  points  K  and  M.  Join  K  and 


FIG.  18. 

M  to  C,  determining  upon  AK  and  AM  the  points  L  and  N 
respectively.  The  diagonal  LN  of  the  quadrangle  KLMN  bisects 
the  segment  AC,  since  LANG  is  a  parallelogram.  Hence:  //  D 
is  the  ideal  point  on  the  point-row  AC,  its  harmonic  conjugate  with 

respect  to  A  and  C  bisects  the 
segment  AC. 

27.  Normal  Conjugate 
Rays  in  a  Harmonic  Pencil. 
— Let  abed  be  a  harmonic 
pencil  of  rays  in  which  the 
conjugate  rays  b  and  d  are 
normal,  i.e.,  perpendicular, 
to  each  other  (Fig.  19).  If 
we  cut  this  pencil  by  a  line 
perpendicular  to  b,  and, 

therefore,  parallel  to  d,  we  obtain  a  harmonic  range  (theorem  IV) 
one  of  whose  points,  viz.,  D,  is  ideal.  Consequently  B  bisects  AC, 
b  bisects  the  angle  (ac),  and  d  bisects  the  supplement  of  the  angle 
(ac).  Hence: 


HARMONIC  RANGES  AND  PENCILS  31 

//  two  conjugate  rays  of  a  harmonic  pencil  are  normal,  to  each 
other,  they  bisect  the  angles  formed  by  the  other  pair  of  conjugate 
rays. 

Exercises  , 

1.  State  and  prove  the  converse  of  the  theorem  in  Art.  26. 

2.  State  and  prove  the  converse  of  the  theorem  in  Art.  27. 

3.  Bisect  a  given  segment  AC,  having  given  a  parallel  to  it. 
Suggestion. — Project  A  and  C  from  a  point  L,  determining  upon  the 

given  parallel  the  points  K  and  M  respectively.  Let  KC  and  MA 
meet  in  N.  The  line  LN  bisects  AC. 

4.  Draw  a  parallel  to  a  given  line  AC  through  a  given  point  K, 
knowing  the  middle  point  of  the  segment  AC. 

Suggestion. — Let  B  be  the  middle  point.  Join  K  to  A  and  C  and 
through  B  draw  any  line  meeting  KA  and  KC  in  L  and  N  respectively. 
Join  L  to  C  and  N  to  A  determining  a  point  M.  The  line  KM  is 
parallel  to  AC. 

5.  In  a  plane  are  given  a  parallelogram  and  a  segment  AC  of  a 
straight  line;  it  is  required,  without  the  use  of  circles,  to  bisect  AC 
and  to  draw  a  parallel  to  AC',  also  to  divide  AC  into  n  equal  parts. 

6.  Through  a  given  point  P  draw  a  straight  line  meeting  two  given 
lines  of  the  plane  in  A  and  B  so  that  (1)  the  segment  AB  shall  be 
bisected  at  P,  (2)  the  segment  AP  shall  be  bisected  at  B.     Under 
what  circumstances  is  the  solution  impossible? 

7.  A  given  straight  line  intersects  the  sides  of  a  triangle  ABC 
in  the  points  AiBiCi.     If  the  harmonic  conjugate  of  each  of  these 
points  with  respect  to  the  two  vertices  on  the  same  side  is  joined  to 
the  opposite  vertex,  show  that  the  three  lines  thus  obtained  meet  in 
a  point  (cf.  Art.  19,  exercise  4). 

8.  If  A  BCD  is  a  harmonic  range  and  a  circle  is  described  upon  AC 
as  diameter,  of  which  S  is  any  point,  prove  that  the  arc  subtending 
the  angle  BSD,  or  its  supplement,  is  bisected  at  A  or  at  C. 

28.  Cross-ratio  of  a  Harmonic  Range. — If  ABCD  is  any  range 
of  points  along  a  line,  then  the  numerical  value  of  the  expression 

AB      AD 
BC  *  DC 

determined  from  the  lengths  and  directions  of  the  several  seg- 
ments involved,  is  called  the  cross-ratio  or  anharmonic  ratio  of 
the  range. 


32  PROJECTIVE  GEOMETRY  [§29 

Let  A  BCD  be  a  harmonic  range.  Project  this  range  from  any 
point  S  (Fig,  20)  and  cut  the  resulting  pencil  by  a  line  through 
B  parallel  to  SD.  The  resulting  range  AiBCi  °°  is  harmonic 
(theorem  IV).  By  similar  triangles, 

AB  _AD 

AtB  ~  SD 
and 

BC  'CD 

Bd  ~  SD 
Since  AiB  —  BC\  (Art.  26),  we  obtain  by  division, 

AB_AJ)_       AD 

BC  ~  CD  "       DC' 

Hence:    The  cross-ratio  of  a  harmonic  range  is  —  1. 


29.  Harmonic  Mean  Between  Two  Numbers. — If  three  numbers 
are  in  arithmetic  progression,  their  reciprocals  are  said  to  be  in 
harmonic  progression.  For  example,  }£,  %,  K  are  in  harmonic 
progression,  since  3,  5,  7  are  three  numbers  of  an  arithmetic  pro- 
gression. In  general,  I/a,  1/6,  1/c  are  in  harmonic  progression  if 
a,  b,  c  are  three  numbers  of  an  arithmetic  progression.  Thus  l/b 
is  the  harmonic  mean  between  I/a  and  1/c  if  b  is  the  arithmetic 
mean  (average)  between  a  and  c. 

If  A  BCD  is  a  harmonic  range  of  points,  then  (Art.  28) 

AB      AD  AB  AD 


~-R  or 


BC      CD       AC  -AB      AD-  AC 


J30]     .   •        HARMONIC  RANGES  AND  PENCILS  33 

from  which  we  easily  find  that 

1111 
AB     AC~AC     AD 

Therefore  the  three  fractions 

l/AB,  I/AC,  I/ AD 

are  in  arithmetic  progression  and  consequently  the  segments 
AB,  AC,  AD 

are  in  harmonic  progression. 

Hence;  //  A  BCD  is  a  harmonic  range,  the  segment  AC  is  the  har- 
monic mean  between  the  segments  AB  and  AD. 

Exercises 

1.  The  points  ABC  are  situated  along  a  line  at  distances  of  one, 
three,  and  four  units  from  0,  respectively.     How  far  from  0  is  the 
harmonic  conjugate  of  each  point  with  respect  to  the  other  two? 

2.  Three  points  A  BC  are  situated  along  a  line  so  that  B  is  two  units 
from  A,  and  C  is  three  units  from  A.     Find  the  position  of  a  series  of 
points,  DEFGH  .    .    .,  so  that  ABCD,  ACDE,  ADEF,  AEFG,  .    .    . 
are  harmonic  ranges.     The  scale  thus  constructed  is  called  a  harmonic 
scale. 

30.  Geometric  Mean  Between  Two  Numbers. — If  a,  b,  c  are 

three  numbers  in  geometric  progression,  then  b  is  the  geometric 
mean  between  a  and  c,  or  b2  —  ac. 

Let  ABCD  be  a  harmonic  range  and  M  the  mid-point  of  the 
segment  BD  (Fig.  21).     Since 

AB      AD 
BC  ~  CD' 
we  have, 

AM  -EM  _  AM  +  BM 
BM  -  CM  ~  BM  +  CM 
Clearing  of  fractions  and  reducing,  we  get 
5M2=  AM -CM. 

Hence:  //  ABCD  is  a  harmonic  range  and  M  is  the  mid-point  of 
the  segment  BD,  then  BM  is  the  geometric  mean  between  AM  and 
CM. 

3 


34  PROJECTIVE  GEOMETRY  [§31 


FIG.  21. 

31.  The  Circle  of  Apollonius. — If  we  project  the  harmonic 
range  A  BCD  from  a  point  K  on  the  circle  whose  diameter  is  BD 
(Fig.  21),  we  obtain  a  harmonic  pencil  having  two  normal  conju- 
gate rays,  KB  and  KD.     Hence  (Art.  27)  KB  bisects  the  angle 
AKC,  and  therefore 

AK  ^AB 

KG  ~  EC 

The  ratio  AK  :KC  is  therefore  constant  so  long  as  K  is  on  the 
circle.  In  other  words:  The  locus  of  a  point,  the  ratio  of  whose 
distances  from  two  fixed  points  is  a  given  constant,  is  a  circle.  This 
circle  is  known  as  the  circle  of  Apollonius. 

32.  Orthogonal   Circles. — Let   ABCD   be-  a   harmonic   range 
(Fig.  22).     If  through  A  and  C  we  draw  any  circle  and  from  M, 
the  center  of  the  circle  of  Apollonius,  the  two  tangents  MT  and 
MTi,  we  have  from  plane  geometry, 

MT2  =  Jfr?  =  AC-CM. 
But  (Art.  30), 

MB*  =  AC -CM. 
Therefore, 

MT  =  MT,  =  MB. 

Consequently  the  points  of  contact  of  these  tangents  lie  on  the 
circle  of  Apollonius. 

Again,  if  0  is  the  center  of  the  circle  through  A  and  C,  the  angle 
OTM  =  the  angle  07\M  =  90°. 

Therefore:  Any  circle  through  A  and  C  cuts  the  circle  of  Apollonius 
at  right  angles. 


§32] 


HARMONIC  RANGES  AND  PENCILS 


35 


The  totality  of  circles  that  can  be  drawn  through  A  and  C  forms 
a  system  of  circles  orthogonal  to  the  circle  of  Apollonius. 


FIG.  22. 


Exercises 

1.  Given  two  pairs  of  points,  AB  and  CD,  upon  the  same  straight 
line  which  do  not  separate  each  other.     With  the  aid  of  circles  find 
two  points  which  harmonically  separate  each  pair. 

Suggestion. — Choose  any  point,  P,  not  on  the  line  containing  the 
given  pairs.  Draw  the  circles  PAB  and  PCD  intersecting  again  in 
the  point  E.  Let  the  straight  line  PE  meet  AB  in  0.  Draw  a  tangent 
from  0  to  PAB  or  to  PCD  and  let  the  point  of  contact  be  T.  The 
circle  whose  center  is  O  and  whose  radius  is  OT  will  cut  the  line  AB 
in  the  required  points  (cf.  Art.  32). 

2.  Given  two  pairs  of  points,  AB  and  CD,  upon  the  same  straight 
line.     Find  the  locus  of  points  from  which  the  segments  A  B  and  CD 
subtend  equal  angles. 

Suggestion. — By  the  preceding  exercise,  find  the  two  points,  M 
and  N,  which  separate  harmonically  each  of  the  pairs  AD  and  BC. 
From  any  point  of  the  sphere  whose  diameter  is  MN,  the  segments 
AB  and  CD  subtend  equal  angles  (Art.  27). 

3.  If  two  pairs  of  points,  AB  and  CD,  upon  the  same  straight  line 
separate  each  other,  find  the  locus  of  points  from  which  AB  and  CD 
subtend  equal  angles. 

4.  If  A  BCD  is  a  harmonic  range,  show  that 

J L  +  _L. 

AC      AB^AD 


36  PROJECTIVE  GEOMETRY  [§32 

6.  If  abed  is  a  harmonic  pencil  of  rays  and  (ab)  indicates  the  angle 
which  a  makes  with  b,  show  that 

sin(ab)  _        sin  (ad) 
sin(6c)   ~        sin(dc) 

6.  If  ABCD  is  any  range  of  points  along  a  straight  line,  show  that 

AB-CD  +  AC-DB  +  AD-EC  =  0. 

7.  If  ABCD  is  a  harmonic  range,  show  that  the  cross-ratio  of  the 
range  ACBD  is  2;  of  ADBC  is  Y^. 

NOTES. — The  idea  of  four  harmonic  points,  or  harmonic  division, 
was  known  to  the  early  Greek  geometers,  but  who  first  invented  it 
is  not  definitely  known.  Apollonius  of  Perga  (247  B.C.)  mentions 
it  in  his  book  on  conic  sections. 

The  harmonic  property  of  a  complete  quadrangle  is  contained  in 
the  Collections  of  Pappus  (300  A.D.).  It  was  made  the  foundation 
for  Von  Staudt's  Geometric  der  Lage,  1847. 

Three  cords  consisting  of  the  same  substance  and  having  the  same 
size  and  tension,  and  whose  lengths  are  in  harmonic  progression,  will 
vibrate  in  harmony  when  struck  in  unison.  The  name  harmonic  is 
probably  due  to  this  fact. 

The  theory  of  cross-ratios  is  due  to  Mobius  (Der  Barycentrische 
Calcul,  1827),  to  Steiner  (Systematische Entwickelung  .  .  .,1832),  and 
to  Chasles  (Aperc.u  Historique  .  .  .,1837).  Mobius  called  the  cross- 
ratio  of  four  points  the  "double-ratio;"  Chasles  called  it  "anharmonic 
ratio."  Some  properties  of  the  cross-ratio  were  known  much  earlier 
and  are  to  be  found  in  the  Collections  of  Pappus. 


CHAPTER  V 

PROJECTIVELY  RELATED   PRIMITIVE   FORMS   OF  THE 
FIRST  KIND 

.  33.  Primitive  Forms  of  the  First  Kind. — The  point-row,  the 
sheaf  of  rays,  and  the  sheaf  of  planes  can  be  correlated,  one  with 
another,  by  the  operations  of  projection  and  section  (Art.  15). 
Moreover,  no  one  of  the  remaining  eight  primitive  forms  can  be 
derived  by  projection  or  section  from  any  one  of  these  three. 
Also  the  dual,  or  reciprocal,  of  any  one  of  these  three  primitive 
forms  is  again  one  of  the  same  three  forms,  and  is  never  one  of 
the  remaining  eight. 

The  point-row,  the  sheaf  of  rays,  and  the  sheaf  of  planes  thus 
constitute  a  group  by  themselves  and  are  called  primitive  forms 
of  the  first  kind.  Primitive  forms  of  the  first  kind  are  one 
dimensional  forms  (Art.  15). 

34.  Chains  of  Perspectivity. — A  series  of  primitive  forms  so 
arranged  that  any  one  of  them  is  in  perspective  position  with  the 
next  in  order  constitutes  a  chain  of  perspectivity.  Thus, 

u  7\  S  A"  Si  X  u\  X  Sz 

is  a  chain  of  perspectivity. 

The  first  and  the  last  primitive  form  of  any  chain  of  perspec- 
tivity are  not  in  general  in  perspective  position  with  each  other, 
but  they  are  correlated,  element  to  element,  and  by  virtue  of 
theorem  IV,  we  can  say  that:  to  any  four  harmonic  elements  of 
either  there  always  correspond  four  harmonic  elements  of  the  other. 

Exercises 

1.  If  u  ^  S  ^  Ui,  are  u  and  MI  in  perspective  position? 

2.  Construct  a  figure  illustrating  the  statement 

u  A  S  A  Si  ^  ui. 

If  M  is  any  point  on  u,  construct  the  corresponding  point  on  u\. 

37 


38 


PROJECTIVE  GEOMETRY 


(§35 


3.  A  point  P  is  joined  to  the  vertices  of  a  triangle  ABC,  the  joining 
lines  meeting  the  sides  opposite  A,  B,  C  in  D\,  E\,  F\  respectively. 
The  two  triangles  ABC  and  DiEiFi  are  in  perspective  position. 
Why?  The  axis  of  perspectivity  meets  the  sides  of  the  triangle  ABC 
in  the  points  D,  E,  F,  the  harmonic  conjugates  of  D\,  Ei,  FI,  with 
respect  to  the  two  vertices  of  ABC  on  the  same  side.  Why?  The 
axis  meets  the  lines  PA,  PB,  PC  in  the  points  R,  S,  T,  respectively. 
Prove  that  the  ranges  FRED,  FSDE,  and  ETDF  are  harmonic. 
Also  the  ranges  RTSF,  SRTD,  and  RSTE  are  harmonic. 

35.  Definition  of  Projective  Relationship. — Two  correlated 
primitive  forms  of  the  first  kind  are  projectively  related  if  to  any 
four  harmonic  elements  of  either  there  correspond  four  harmonic 
elements  of  the  other.  Thus,  if  two  primitive  forms  are  in  per- 


A  M  L   K    B        C  D  E 

FIG.  23. 

spective  position,  or  if  they  belong  to  a  chain  of  perspectivity, 
they  are  projectively  related  (theorem  IV). 

The  symbol  X  placed  between  two  primitive  forms  indicates 
that  they  are  projectively  related.  Thus, 

S  A  Si 

means  that  the  two  sheaves  of  rays  whose  centers  are  S  and  Si 
are  projectively  related. 

When  two  primitive  forms  are  projectively  related,  we  shall 
say  that  a  projectivity  exists  between  them.  For  example,  a 
projectivity  exists  between  the  first  primitive  form  and  the  last, 
in  any  chain  of  perspectivity. 

36.  Harmonic  Scales. — With  three  points  A,  B,  C  of  a  point- 
row,  chosen  arbitrarily,  we  can  determine  a  series  of  points  D,  E, 


§37]          PRIMITIVE  FORMS  OF  THE  FIRST  KIND  39 

F,G,  .  .  .  ,  M,L,  K  (Fig.  23)  such  that  ABCD,  ACDE,  ADEF, 
.  .  .  ,  AMLK,  ALKB,  AKBC  are  harmonic  ranges.  The  series 
of  points  thus  constructed  is  a  harmonic  scale.  Each  point  of 
the  scale  is  harmonically  separated  from  A  by  the  two  points 
which  stand  next  to  it.  For  example,  L  is  harmonically  sepa- 
rated from  A  by  M  and  K.  A  is  called  the  origin  of  the  scale. 

The  projector  of  a  harmonic  scale  on  a  point-row  is  a  harmonic 
scale  in  a  sheaf  of  rays;  the  projector  of  a  harmonic  scale  in  a 
sheaf  of  rays  is  a  harmonic  scale  in  a  sheaf  of  planes.  It  follows, 
from  theorem  IV,  that  any  section  of  a  harmonic  scale  in  a  sheaf 
of  rays  (sheaf  of  planes)  is  a  harmonic  scale  on  a  point-row  (in  a 
sheaf  of  rays). 

If  two  point-rows  are  protectively  related  and  we  know  three 
pairs  of  corresponding  points,  we  can  construct  infinitely  many 
pairs  of  corresponding  points.  For  if  A,  B,  and  C  correspond 
respectively  to  AI,  B\,  and  C\,  the  harmonic  scales  determined 
by  ABC  and  A\BiC\  must  correspond,  point  to  point,  in  the  given 
projectivity  (Art.  35). 

Exercises 

1.  If  A,  B,  and  C  are  three  points  along  a  straight  line,  B  being 
three  units  from  A  and  C  four  units  from  A,  construct  the  harmonic 
scale  determined  by  A,  B,  and  C. 

2.  Given  any  three  points  A,  B,  C  of  a,  point-row,  construct  the 
harmonic  scale  determined  by  them. 

3.  If  the  origin  of  a  harmonic  scale  is  the  ideal  point  on  the  point- 
row,  show  that  the  points  of  the  scale  are  equidistant  from  each  other. 

4.  In  exercise  1  find  the  cross-ratio  of  the  range  BCDE. 

37.  Two  Pairs  of  Points  Each  Harmonically  Separated  by  a 
Third  Pair. — Theorem  V. — //  a  pair  of  points  AC  harmonically 
separates  each  of  two  other  pairs  of  points,  BD  and  B\D\,  then  the 
pair  BD  is  not  separated  by  the  pair  BtDi. 

Let  ABCD  (Fig.  24)  be  a  harmonic  range  and  KLMN  a  quad- 
rangle defining  it.  Suppose  the  points  A,  K,  L,  and  C  remain 
fixed  while  M  and  N  move  along  the  fixed  lines  LC  and  KG  re- 
spectively. The  two  variable  points  B  and  D  will  always  be 
harmonically  separated  by  A  and  C.  To  every  position  of  B 
there  is  just  one  position  of  D  (theorem  III).  When  B  moves  in 
one  direction  along  the  line  AC,  D  moves  in  the  opposite  direc- 


40 


PROJECTIVE  GEOMETRY 


[§38 


Bl      C  Di 

FIG.  24. 


tion.  As  B  describes  the  finite  segment  CA,  D  describes  the 
infinite  segment  CA.  Consequently,  no  two  pairs,  as  BD  and 
BiDi,  can  ever  separate  each  other. 

38.  Converse  of  Theorem  V. — //  two  pairs  of  points,  BD  and 
BiDi,  do  not  separate  each  other,  then  there  always  exists  at  least 
one  pair  of  points  AC  which  harmonically  separates  each  of  the 
others. 

Since  B  and  D  are  not  separated  by  BI  and  D\,  it  follows  that 
both  BI  and  DI  are  on  one  of  the  segments  formed  by  the  pair 
B,  D  (Fig.  25). 

Let  A  be  a  point  on  the  segment  BD  not  occupied  by  B\  and  DI. 
Construct  C\,  the  harmonic  conjugate  of  A  with  respect  to  B  and 


Co 


D 


FIG.  25. 


D,  and  also  C?,  the  harmonic  conjugate  of  A  with  respect  to  BI 
and  DI.  By  theorem  V,  as  A  describes  the  segment  BD  upon 
which  it  lies,  d  will  describe  the  complementary  segment  BD 
and  (7  2  will  remain  upon  the  segment  fiiDj  not  occupied  by  A. 
Consequently  C\  and  Cz  must  eventually  coincide  in  at  least  one 
point  C  on  the  segment  B\D\  not  occupied  by  A,  and  then  A  and 
C  harmonically  separate  both  pairs,  BD  and  B\D\  (cf.  Art.  32, 
exercise  1). 


§39] 


PRIMITIVE  FORMS  OF  THE  FIRST  KIND 


41 


Exercises 

1.  If  .A,  B,  C,  D,  are  four  points  on  a  line  such  that  A,  B  is  not 
separated  by  C,  D,  nor  A,  D,  by  B,  C,  construct  the  pair  of  points 
harmonically  separating  A,  B  and  C,  D;  also  the  pair  harmonically 
separating  A,  D  and  B,  C  (cf.  Art.  32,  exercise  1). 

2.  By  means  of  projection,  show  that  theorem  V  and  its  converse 
apply  to  pairs  of  rays  in  a  sheaf  of  rays,  and  to  pairs  of  planes  in  a 
sheaf  of  planes. 

39.  Consequences  of  Theorem  V  and  Its  Converse.— With  the 
aid  of  theorem  V  and  its  converse,  we  can  reach  some  important 
conclusions  concerning  projectively  related  primitive  forms  of  the 
first  kind.  For  example,  suppose 
u  and  MI  are  two  projectively  re- 
lated point-rows,  the  points  A,  B, 
C,Dofu  corresponding  respectively 
to  the  points  A\,  B\,  C\,  DI  of  u\. 
Then,  if  A,  B  does  not  separate  C, 
D,  it  follows  that  A\,  B\  cannot 
separate  C\,  DI.  For  we  can  find 
two  points  M  and  N  which  har- 
monically separate  A,  B  and  C,  D 
(converse  of  theorem  V).  The 
ranges  A  MBN  and  MCND  are  then 
harmonic  and,  by  the  definition  of  projective  relationship,  corre- 
spond to  harmonic  ranges  on  u\.  Thus,  if  MI,  NI  are  the  points 
corresponding  to  M,  N,  then  A\M \B\Ni  and  M\C '\N\D\  are  har- 
monic ranges.  Consequently  A\,  B\  cannot  separate  C\,  D\ 
(theorem  V). 

The  property  just  proved  for  projectively  related  point-rows 
can  be  extended  at  once  to  any  two  projectively  related  primitive 
forms  of  the  first  kind  by  means  of  projection.  In  general,  there- 
fore; If  two  primitive  forms  of  the  first  kind  are  projectively  related, 
any  two  non-separating  pairs  of  elements  of  the  one  always  correspond 
to  two  non-separating  pairs  of  elements  of  the  other. 

Returning  to  the  case  of  two  projectively  related  point-rows, 
we  can  say  that  any  point  P  on  the  segment  BC  not  occupied  by 
A,  D  (Fig.  26),  corresponds  to^-a  point  Pi  on  the  segment  B\C\  not 
occupied  by  AI,  DI.  For  A,  B  and  P,  C,  and  also  B,  P  and  C,  D, 


FIG.  26. 


42  PROJECTIVE  GEOMETRY  [§39 

are  pairs  of  non-separating  points,  and  hence  AI,  B\  cannot 
separate  PI,  C\,  nor  can  d,  D\  separate  B\,  PI. 

The  operation  of  interpolating  points  on  any  segment  of  u  can 
be  continued  indefinitely,  and  the  corresponding  points  must  occur 
on  u\  in  exactly  the  same  order  as  they  are  placed  upo'n  u.  Thus 
the  arrangement  ABPQ CD  must  correspond  to  the  arrange- 
ment AiBiPiQi CiDi.  This  result  is  evidently  immediately 

carried  over  to  the  other  primitive  forms  of  the  first  kind  by  means 
of  projection. 

If  we  commence  with  three  points  A,  B,  C  upon  u,  we  can  con- 
struct by  repeated  use  of  harmonic  ranges  an  indefinitely  great 
number  of  points -upon  u.  Theorem  V  and  its  converse  enable  us 
to  say  that  the  corresponding  points  upon  the  projectively  related 
point-row  u\  must  occur  in  the  same  order  as  they  occur  upon  u. 

But  it  has  been  shown1  that  construction  by  harmonic  ranges, 
even  if  repeated  an  infinite  number  of  times,  can  never  construct 
all  the  points  upon  u.  Theoretically,  we  may  arrive  at  a  point- 
row  whose  points  are  everywhere  dense;  that  is,  such  that  be- 
tween any  two  of  its  points  there  are  an  infinite  number  of  con- 
structed points.  But  the  point-row  constructed  in  this  way  is 
not  continuous.  A  familiar  example  from  metric  geometry  may 
serve  to  illustrate.  Suppose  that  B  is  one  unit  from  A  and  we 
interpolate  upon  the  segment  AB  all  the  points  whose  distances 
from  A  are  proper  fractions.  The  constructed  points  are  every- 
where dense,  but  the  point  whose  distance  from  A  is  H\/2  is  on 
the  segment  and  is  not  one  of  the  constructed  points. 

From  what  has  been  said,  it  appears  that  theorem  V  and  its 
converse  enable  us  to  say  that,  if  u/\Ui,  a  point-row  upon  u 
everywhere  dense  corresponds  to  a  point-row  upon  Ui  also  every- 
where dense,  the  arrangement  of  points  upon  the  two  point-rows 
being  the  same. 

To  arrive  at  continuously  projective  point-rows  use  is  made  of 
the  Dedekind  postulate,  which  can  be  stated  as  follows: 

//  the  infinite  succession  of  points  APQR STB  on  any  seg- 
ment AB  of  u,  constructed  by  repeated  use  of  harmonic  ranges,  is 
divided  into  two  sets  such  that: 

1  The  proof  is  beyond  the  scope  of  this  book*  Cf.  Clebsch-Lindemann,  Vorlesungen 
liber  Geometrie,  vol.  ii,  p.  433,  where  other  references  are  given. 


§39]          PRIMITIVE  FORMS  OF  THE  FIRST  KIND  43 

First, — any  point  of  the  succession  A B  belongs  to  one  of 

the  two  sets; 

Second, — the  terminal  A  belongs  to  the  first  set  and  B  to  the  second; 

Third, — any  point  whatever  of  the  first  set  precedes  every  point 
of  the  second: 
then  there  exists  one,  and  but  one,  point  M  such  that  all  points  of 

the  succession  A B  that  precede  M  belong  to  the  first  set,  and 

all  points  of  the  succession  A B  that  succeed  M  belong  to  the 

second  set. 

The  point  M,  whose  existence  is  thus  postulated,  may  belong 
to  the  first  set  in  which  case  M  is  the  last  point  of  the  set,  or  M 
may  belong  to  the  second  set  in  which  case  it  is  the  first  point  of 
the  second  set,  or  M  may  belong  to  neither  set  in  which  case  the 
first  set  has  no  last  point  and  the  second  set  has  no  first  point. 
In  this  last  case  M  is  a  new  point  of  the  segment  AB,  not  belong- 
ing to  the  succession  A B  of  constructed  points,  and  is  de- 
fined by  the  two  sets  of  points  into  which  the  succession  A B  is 

divided. 

The  division  of  the  succession  A B  into  two  sets  of  points 

effects  a  corresponding  division  in  the  succession  AiPiQiRi 

SiTiBi  on  the  projectively  related  point-row  MI,  and  this  division 
defines  the  point  MI  corresponding  to  M. 

The  Dedekind  postulate  thus  enables  us  to  add  a  series  of  new 
points  on  AB  not  constructible  by  means  of  harmonic  ranges; 
and  a  repeated  application  of  this  postulate  to  the  succession  of 
points  already  constructed  makes  it  theoretically  possible  to  reach 
all  the  points  on  the  segment  AB  and  consequently  all  the  points 
on  the  line  u,  since  A  B  is  any  segment  of  u.  To  each  point  of  u 
there  is  a  corresponding  point  on  u\  arrived  at  by  corresponding 
operations  upon  the  constructed  points  of  u\. 

Clearly,  what  has  been  said  about  the  points  of  u  and  u\  is 
immediately  carried  over  by  projection  to  sheaves  of  rays  and 
sheaves  of  planes,  and  we  are  led,  finally,  to  the  important 
continuity  theorem. 

Theorem  VI. — //  a  moving  element  describes  continuously  either 
of  two  projectively  related  primitive  forms  of  the  first  kind,  the 
corresponding  element  must  describe  continuously  the  other. 


44 


PROJECTIVE  GEOMETRY 


Exercise 


(§40 


In  Fig.  27,  u^S  /\Ui  A#I  A  "2  and  therefore  wA«2.  Note  that  any 
point  on  the  finite  segment  A  B  corresponds  to  a  point  on  the  infinite 
segment  A zB2,  and  vice  versa.  If  M  is  the  ideal  point  on  u,  construct 
MZ]  if  Nt  is  the  ideal  point  on  u2,  construct  N. 

40.  Superposition. — Self-corresponding  Elements. — Two  point- 
rows  are  superposed  when  they  lie  upon  the  same  straight  line; 
two  sheaves  of  rays  are  superposed  when  they  are  coplanar  and 
concentric;  two  sheaves  of  planes  are  superposed  when  they  have 
a  common  axis,  or  are  coaxial. 

Two  protectively  related  primitive  forms  of  the  first  kind,  con- 
sisting of  like  elements,  may  be  compared  by  superposition.  To 


B      A 


FIG.  27. 

compare  primitive  forms  in  this  way  is  the  same  kind  of  operation 
as  that  employed  by  the  draughtsman  when  he  compares  his 
scale  with  a  line  to  be  measured,  or  his  protractor  with  an  angle 
to  be  measured. 

When  two  protectively  related  primitive  forms  are  superposed, 
it  may  happen  that  some  elements  of  one  of  them  coincide  with 
their  corresponding  elements  of  the  other.  Such  elements,  if  they 
exist,  are  called  self-corresponding  elements,  or  double  elements. 
The  possibility  of  the  existence  of  at  least  one  self-corresponding 
element  is  evident,  for  two  projectively  related  primitive  forms 
can  be  superposed  in  such  a  way  that  any  particular  element  of 


§40]          PRIMITIVE  FORMS  OF  THE  FIRST  KIND  45 

one  falls  upon  its  corresponding  element  of  the  other.  This  is 
actually  done,  for  example,  when  two  scales  are  compared  by 
placing  their  zero  points  together.  In  certain  cases,  however, 
it  is  easy  to  see  that  there  must  be  at  least  two  self-corresponding 
elements.  If,  for  example,  u  and  u\  (Fig.  28)  are  two  protectively 
related  point-rows,  superposed  so  that  the  arrangement  of  points 
ABC-  —  runs  from  left  to  right  while  the  corresponding  arrange- 


FIG.  28. 

ment  A  \B\Ci runs  from  right  to  left,  then  it  is  evident  that 

a  variable  point  which  describes  one  of  them  continuously  must 
meet  and  pass  its  corresponding  point  at  least  twice.  Thus,  in 
this  case,  there  must  be  at  least  two  self-corresponding  points. 
The  point-rows  themselves  are  said  to  be  oppositely  projective. 


FIG.  29. 

If  the  point-rows  are  directly  projective ;  that  is,  if  correspond- 
ing arrangements  of  points  run  in  the  same  direction,  there  are 
not  necessarily  any  self-corresponding  points.  If,  however,  any 
segment  AB  of  u  (Fig.  29)  is  included  in  the  corresponding  seg- 
ment A iBi  of  Ui  (or  vice  versa),  it  is  easy  to  see  that  there  must  be 
at  least  one  self-corresponding  point. 

Figs.  28  and  29  illustrate  the  two  cases  for  protectively  related 
and_superposed  sheaves  of  rays. 


46  PROJECTIVE  GEOMETRY  [§41 

t 
41.  Von  Staudt's  Fundamental  Theorem. — Theorem  VII. — // 

two  projectively  related  primitive  forms  of  the  first  kind  are  super- 
posed and  have  three  self-corresponding  elements,  then  all  their 
elements  are  self-corresponding  and  the  two  forms  are  consequently 
identical.1 

Consider  at  first  two  projectively  related  and  superposed  point- 
rows  u  and  MI,  and  let  A,  B,  and  C  be  the  three  self-corresponding 
points,  so  that  A\,  BI,  C\  fall  respectively  at  A,  B,  C  (Fig.  30). 
The  harmonic  scales  determined  by  ABC  and  by  AiBiCi  must 
coincide,  point  for  point,  by  virtue  of  the  projective  relationship. 
There  is,  then,  an  infinity  of  self-corresponding  points.  But 
this  does  not  exclude  the  possibility  that  some  point,  not  belong- 
ing to  the  harmonic  scale  determined  by  ABC,  fails  to  coincide 
with  its  corresponding  point.  Suppose,  for  example,  that  a 
point  P  on  the  finite  segment  AB  does  not  coincide  with  its  corre- 
sponding point  PI.  Since  the  arrangement  APBC  must  corre- 

A     N          p  M  B  C        u 

•e — *— e o   x 


B1  Ci        MI 

FIG.  30. 

spond  to  the  arrangement  A\PiB\Ci  (Art.  39),  it  follows  that  P\ 
must  lie  on  the  same  segment  with  P.  As  P  moves  continuously 
towards  B,  Pi  must  move  continuously  toward  B  by  virtue  of  the 
continuity  theorem.  P  and  PI  must  coincide,  therefore,  either  at, 
or  before,  reaching  B.  Let  us  suppose  they  first  coincide  at  M. 
Similarly,  in  moving  toward  A,  let  us  suppose  they  first  coincide 
at  N.  We  shall  then  have  a  segment  MN  within  the  finite  segment 
AB,  or  coinciding  with  it,  on  which  no  point  coincides  with  its  cor- 
responding point.  But  this  is  impossible,  since  the  harmonic  con- 
jugate of  C  with  respect  to  M  and  N  lies  on  the  finite  segment 
MN  and  coincides  with  its  corresponding  point  by  virtue  of  the 
projectivity  between  u  and  MI.  Consequently,  every  point  is  a 
self-corresponding  point  and  the  two  point-rows  are  identical. 

The  same  argument  obviously  holds  for  projectively  related  and 
superposed  sheaves  of  rays  or  sheaves  of  planes.  Or  we  may 
prove  the  theorem  for  these  primitive  forms  by  means  of  projection, 
having  established  it  for  superposed  point-rows. 

1  Theorem  VIII  (Art.  43)  is  often  called  Von  Staudt's  fundamental  theorem. 


§42] 


PRIMITIVE  FORMS  OF  THE  FIRST  KIND 


47 


Theorem  VII  is  of  great  importance  in  the  further  development 
of  pure  projective  geometry.  It  is  known  as  the  fundamental 
theorem  of  Von  Staudt. 

42.  Consequences  of  Theorem  VII. — The  principal  results  that 
depend  upon  theorem  VII  are  stated  in  the  following  corollaries. 

Corollary  I.—//  two  distinct  primitive  forms  of  the  first  kind  are 
protectively  related  and  superposed,  they  cannot  have  more  than  two 
self-corresponding  elements. 

For,  if  they  have  three  self-corresponding  elements,  they  are 
identical. 


Corollary  2. — //  two  primitive  forms  of  the  first  kind  consisting 
of  unlike  elements  are  projectively  related  and  three  elements  of  the 
first  lie  upon  their  corresponding  elements  of  the  second,  then  the 
two  forms  are  in  perspective  position. 

Suppose   u(ABC )  X  S(abc )    (Fig.  31),'  the  points  A, 

B,  C  lying  on  their  corresponding  rays  a,  b,  c.  The  line  u  cuts  S 
in  the  point-row  ui(AiBiCiDi ).  Hence,  we  have 

UI~K  s  A  u. 

Therefore,  to  any  harmonic  range  on  u\  there  corresponds  a  har- 
monic range  on  u,  and  consequently  u  and  u\  are  projectively 
related,  superposed,  and  have  three  self-corresponding  points; 
namely,  A,  B,  and  C.  The  point-rows  u  and  u\.  are  then  identical 
and  S  X  u. 

Exercise 

If  a  sheaf  of  rays  is  projectively  related  to  a  sheaf  of  planes  and  three 
rays  of  the  one  lie  in  their  corresponding  planes  of  the  other,  prove  that 
the  two  primitive  forms  are  in  perspective  position. 


48 


PROJECTIVE  GEOMETRY 


[§42 


Suggestion. — The  plane  of  the  sheaf  of  rays  cuts  the  sheaf  of  planes 
in  a  second  sheaf  of  rays.     How  are  the  two  sheaves  of  rays  related? 


Corollary  3. — If  two  projec- 
tively  related  point-rows  lie  in  the 
same  plane,  but  are  not  super- 
posed, and  have  their  common 
point  as  a  self-corresponding 
point,  then  they  are  in  perspective 
position. 


For  the  corollary  on  the  left,  let  u(ABC— 
A  coinciding  with  AI  (Fig.  32).    Let  the  lines  BB  i 


If  two  protectively  .  related 
sheaves  of  rays  lie  in  the  same 
plane,  but  are  not  concentric,  and 
have  their  common  ray  as  a  self- 
corresponding  ray,  then  they  are 
in  perspective  position. 


and  Cd 


FIG.  33. 

intersect  in  S.  Project  each  point-row  from  S,  thus  obtaining  two 
projectively  related  and  superposed  sheaves  of  rays  having  three 
self-corresponding  rays,  namely,  a,  b,  and  c.  These  sheaves  are 


§42] 


PRIMITIVE  FORMS  OF  THE  FIRST  KIND 


49 


consequently  identical  and  the  two  point-rows  are  thus  sections 
of  the  same  sheaf  of  rays  and  therefore  in  perspective  position. 

Fig.  33  illustrates  the  dual  corollary.     The  reader  can  easily 
supply  the  proof. 


Corollary  4. — //  two  projec- 
tively  related  sheaves  of  rays  are 
not  concentric  and  any  three 
points  of  intersection  of  corre- 
sponding rays  are  collinear,  then 
they  are  in  perspective  position. 


If  two  protectively  related 
point-rows  are  not  superposed 
and  any  three  lines  joining 
corresponding  points  are  con- 
current, then  they  are  in  per- 
spective position. 


For  the  corollary  on  the  left,  cut  each  sheaf  by  the  line  on  which 
lie  the  three  points  of  intersection.  We  thus  obtain  two  projec- 
tively  related  and  superposed  point-rows  having  three  self-corre- 
sponding points.  The  sheaves  of  rays  are  thus  projections  of 
the  same  point-row  and  consequently  in  perspective  position. 

The  reader  should  supply  the  proof  for  the  dual  corollary. 

Corollary  5. — •//  two  protectively  related  sheaves  of  planes  whose 
axes  are  coplanar  have  their  common  plane  as  a  self-corresponding 
plane,  then  they  are  in  perspective  position. 

Since  the  axes  are  coplanar,  they  meet  in  a  point  S.  Cut  the 
two  sheaves  by  any  plane  not  passing  through  S,  and  thus  obtain 
two  protectively  related  sheaves  of  rays  which  are  coplanar  and 
have  their  common  ray  as  a  self-corresponding  ray  (the  trace  of 
the  self-corresponding  plane) .  The  sheaves  of  rays  are,  therefore, 
in  perspective  position  (corollary  3),  and  thus  projectors  of  the 

same  point-row.  Let  u(ABC )  be  this  point-row.  The  two 

sheaves  of  planes  are  projectors  of  the  sheaf  of  rays  S  (ABC ) 

and  consequently  they  are  in  perspective  position. 


Exercises 

1.  Draw  a  figure  illustrating  corollary  5. 

2.  What  is  the  space-dual  of  corollary  5? 

3.  Project  Fig.  32  from  a  point  not  on  the  plane  of  the  two  point- 
rows.     State  and  prove  the  corollary  for  the  two  projectively  related 
sheaves  of  rays  thus  obtained.     In  the  same  way,  project  Fig.  33. 
What  corollary  is  obtained  from  the  resulting  figure? 

4 


50 


PROJECTIVE  GEOMETRY 


[§43 


43.  Determination  of  Projective  Relationship.  —  Theorem  VIII. 
—  A  projectivity  can  always  be  established  between  any  two  primitive 
forms  of  the  first  kind  so  that  any  three  elements  of  the  one  shall 
correspond  to  three  elements  of  the  other  chosen  arbitrarily.  The 
projectivity  thus  established  is  unique. 

Suppose  u  and  HI  are  two  coplanar  point-rows  and  we  choose 
the  points  A,  B,  (7  of  u  to  correspond  respectively  to  A  i,  BI,  dof  MJ 
(Fig.  34).  Join  the  points  of  any  pair,  for  example  B  and  BI, 
and  upon  the  line  BBi  choose  any  two  points  as  S  and  Si.  From 
S  project  ABC  -  and  from  Si  project  A\Bid  -  . 
Then  if 


u(ABC 


we  have 
S(abc 


FIG.  34. 

Hence  S  and  Si  are  projectively  related  and  have  their  common 
ray  (6,  61)  as  a  self-corresponding  ray.  They  are  consequently  in 
perspective  position  (corollary  3).  The  point-row  of  which  both 
S  and  Si  are  projectors  is  constructed  by  joining  the  intersection 
of  a  and  c^to  the  intersection  of  c  and  ci.  If  uz  is  this  point-row, 
then 


To  any  point  as  D  of  u  we  can  now  immediately  construct  the 


§43]          PRIMITIVE  FORMS  OF  THE  FIRST  KIND  51 

corresponding  point  D\  of  u\.     For  the  projectors  of  D  and  D\, 
from  S  and  Si  respectively,  must  intersect  upon  u2. 

That  the  point  Z>i  thus  constructed  is  unique,  is  proved  as 
follows.  If  by  shifting  the  centers  S  and  Si  along  BB\,  or  if  by 
choosing  centers  on  AAi  or  on  CCi,  we  could  arrive  at  a  point 
D'i,  say,  differing  from  D\,  we  would  have 


and  therefore 

A  ^CiDi  --  A-A  iBtdD'i  --  . 

But  this  is  impossible  (theorem  VII).     Consequently  D'i  coincides 
with  Z>i;  or,  in  other  words,  DI  is  uniquely  determined. 

Since  D  is  any  point  on  u,  we  can  as  above  construct  as  many 
pairs  of  corresponding  points  as  we  choose.  The  projectivity  is 
then  completely  determined. 

Having  proved  the  theorem  for  two  coplanar  point-rows,  we 
may  prove  it  for  other  pairs  of  primitive  forms  of  the  first  kind  by 
projection  or  section.  For  example,  two  coplanar  sheaves  of 
rays  in  which  the  rays  abc  are  chosen  to  correspond  to  the  rays 
ai&iCi,  each  to  each,  may  be  cut  by  two  lines  on  which  the  points 
ABC  correspond  to  the  points  AiBiCi,  each  to  each.  The  projec- 
tivity established  between  these  point-rows  as  above,  establishes 
the  projectivity  between  the  sheaves  of  rays.  Again,  two  sheaves 
of  planes  in  which  afty  are  chosen  to  correspond  to  «i/3i7i  can  be 
cut  by  a  plane  in  two  coplanar  sheaves  of  rays  in  which  the  traces 
of  a,  0,  7  correspond  respectively  to  the  traces  of  «i,  0i,  71.  The 
projectivity  established  between  these  sheaves  of  rays  establishes 
the  required  projectivity  between  the  sheaves  of  planes. 
'  On  account  of  its  importance,  we  shall  prove  the  theorem  for 
two  coplanar  sheaves  of  rays  without  reference  to  the  proof  just 
given  for  two  coplanar  point  -rows.  Thus,  (Fig.  35),  if 


and  we  cut  each  sheaf  by  a  line  passing  through  the  intersection  of 
a  pair  of  corresponding  rays  (a  and  0,1  in  the  figure)  we  obtain 
two  point-rows  u(ABC  --  )  and  ui(A\BiC\  --  )  in  perspective 
position  (corollary  3).  We  can  then  immediately  construct  the 


52 


PROJECTIVE  GEOMETRY 


[§43 


ray  di  of  Si  corresponding  to  any  ray  d  of  S.  For,  if  £2  is  the 
center  of  the  sheaf,  of  which  u  and  MI  are  sections,  the  points  (du) 
and  (diUi)  are  collinear  with  S*.  That  d\  is  uniquely  determined 
by  this  process  follows  as  in  the  case  of  two  coplanar  point-rows. 
If  the  process  indicated  could  ever  lead  to  a  ray  d'i  not  coinciding 
with  di,  then  Si  would  be  the  common  center  of  two  projec- 
tively  related  sheaves  of  rays  having  the  three  self-corresponding 
rays  ai&id,  but  in  which  d\  does  not  coincide  with  its  correspond- 
ing ray  d'\,  thus  contradicting  theorem  VII. 

Since  d  is  any  ray  of  S,  the  projectivity  between  S  and  Si  is 
completely  determined. 


FIG.  35. 

The  arguments  set  forth  above  enable  us  to  say  that:  Any  two 
projectively  related  primitive  forms  of  the  first  kind  can  always  be 
connected  by  a  chain  of  perspectivity.  Thus,  in  Fig.  35, 


Conversely;  if  two  primitive  forms  of  the  first  kind  are  connected 
by  a  chain  of  perspectivity,  they  are  projectively  related  (cf.  Art.  34). 

Exercises 

1.  Given  two  coplanar  point-rows  u  and  MI,  determine  the  pro- 
jectivity between  them  so  that  A,  B,  C  of  u  shall  correspond  re- 


§43]          PRIMITIVE  FORMS  OF  THE  FIRST  KIND  53 

spectively  to  Ai,  B\,  Ci  of  MI:  (a)  when  A,  B,  C  and  AI,  BI,  C\,  are 
all  actual  points;  (6)  when  C  and  Ci  are  ideal  points.  Construct  a 
number  of  pairs  of  corresponding  points  in  each  case. 

2.  In  the  preceding  exercise,  construct  the  point  on  each  point-row 
which  corresponds  to  the  point  of  intersection  of  the  point-rows  con- 
sidered as  a  point  on  the  other  point-row.     How  many  lines  joining 
pairs  of  corresponding  points  can  pass  through  any  given  point  in 
the  plane  (corollary  4)? 

3.  Given  two  coplanar  sheaves  of  rays  S  and  Si,  determine  the 
projectivity  between  them  so  that  the  rays  a,  b,  c,  of  S  shall  correspond 
respectively  to  the  rays  ai,  &i,  c\  of  Si.     Construct  a  number  of  pairs 
of  corresponding  rays,  in  particular,  the  ray  in  each  sheaf  correspond- 
ing to  the  ray  SSi  considered  as  a  ray  of  the  other  sheaf. 

4.  How  can  two  protectively  related  sheaves  of  rays  be  placed  in 
perspective  position?     Two  projectively  related  point-rows? 

5.  If  two  sheaves  of  rays  are  coplanar  and  in  perspective  position, 
find  two  rays  in  one  of  them  which  are  perpendicular  to  each  other 
and  which  correspond  to  two  perpendicular  rays  in  the  other.     Show 
that  in  two  projectively  related  sheaves  of  rays,  whose  centers  are  not 
ideal  points,  there  is  always  a  pair  of  homologous  right  angles. 

Suggestion. — If  S/^Si  and  u  is  the  axis  of  perspectivity,  a  circle 
can  be  drawn  through  S  and  Si  whose  center  is  on  u. 

6.  Given  two  fixed  straight  lines  u  and  u\  intersecting  in  0,  and 
two  points  S  and  Si  collinear  with  O.     A  straight  line  rotates  about  a 
fixed  point  U  and  intersects  u  and  HI  in  A  and  AI  respectively. 
Show  that  the  locus  of  the  intersection  of  SA  and  SiAi  is  a  straight 
line  passing  through  0  (Chasles,  Geometric  Superieure,  1880;  also  the 
Collections  of  Pappus).     State  the  dual  proposition. 

7.  If  the  three  sides  of  a  variable  triangle  BC,  AC,  AB  rotate  about 
three  collinear  points  D,  E,  F,  respectively,  while  two  vertices  A  and 
B  move  upon  two  fixed  straight  lines  which  intersect  in  0,  show  that 
the  third  vertex  C  will  describe  a  straight  line  passing  through  0 
(cf.  Desargues  theorem,  Art.  17). 

8.  Prove  the  theorem  stated  in  exercise  4,  Art.  10. 

Suggestion. — The  sheaves  A  (DBF)  and  C  (DBF)  are  in  perspective 
position  and  cut  out  ranges  on  DE  and  FE,  respectively,  that  are  in 
perspective  position  with  each  other  (corollary  3).  L  is  the  center 
of  perspectivity  for  these  ranges. 

9.  If  the  four  vertices,  A,  B,  C,  D  of  &  variable  quadrangle  move 
respectively  upon  four  fixed  lines  which  pass  through  one  point  0, 
while  three  of  the  sides  AB,  BC,  CD,  rotate  about  three  fixed  collinear 


54      _  PROJECTIVE  GEOMETRY  [§43 

points,  then  the  remaining  three  sides  will  also  rotate  about  fixed 
points  (Cremona,  Projective  Geometry,  1885). 

NOTE. — The  concept  of  projective  relationship  between  two  primi- 
tive forms  is  due  to  Mobius  (Der  Barycentrische  Calcul,  1827). 

In  order  to  free  projective  geometry  from  any  considerations  of 
measurement,  Von  Staudt  denned  projective  relationship  as  in  Art. 
34  (Geometric  der  Lage,  1847),  and  based  its  further  development 
upon  the  fundamental  theorem  (theorem  VII)  and  its  consequences. 


CHAPTER  VI 
ELEMENTARY  FORMS 

44.  Definition  of  Elementary  Forms. — If  two  primitive  forms 
of  the  first  kind  consisting  of  like  elements  are  in  perspective 
position,  any  pair  of  corresponding  elements  determines  a  third 
element.  The  locus  of  this  third  element  is  the  primitive  form 
of  which  the  original  two  are  either  projectors  or  sections.  Thus, 
if  two  sheaves  of  rays  are  in  perspective  position,  the  locus  of  the 
intersection  of  corresponding  rays  is  the  point-row  of  which  the 
two  sheaves  of  rays  are  projectors.  Reciprocally,  if  two  point 
rows  are  in  perspective  position,  the  locus  of  the  rays  joining 
corresponding  points  is  the  sheaf  of  rays  of  which  the  two  point- 
rows  are  sections.1 

Two  primitive  forms  in  perspective  position  are  said  to  generate 
the  primitive  form  of  which  they  are  either  projectors  or  sections. 

If  two  primitive  forms  of  the  first  kind  are  projectively  related 
but  not  in  perspective  position  and  any  pair  of  corresponding 
elements  determines  a  third  element  which  varies  with  the  pair, 
the  locus  of  this  third  element  is  called  an  elementary  form.  The 
two  primitive  forms  generate  the  elementary  form.  Thus,  if  two 
sheaves  of  rays  are  coplanar  and  projectively  related,  but  not 
in  perspective  position  nor  concentric,  any  pair  of  corresponding 
rays  determines  a  point.  The  locus  of  this  point  cannot  be  a 
straight  line,  for  then  the  sheaves  of  rays  would  be  projectors 
of  this  line  and  thus  in  perspective  position.  Reciprocally,  if 
two  point-rows  are  coplanar  and  projectively  related,  but  not  in 
perspective  position  nor  superposed,  the  rays  joining  correspond- 
ing points  cannot  all  pass  through  the  same  point,  for  then  the 
point-rows  would  be  in  perspective  position. 

1  If  two  non-intersecting  point-rows  are  in  perspective  position,  they  are  sections 
of  infinitely  many  sheaves  of  planes.  Two  corresponding  points  do  not  determine 
any  plane,  and  the  point-rows  do  not  generate  any  primitive  form.  This  case 
will  be  considered  in  Chapter  X. 

55 


56 


PROJECTIVE  GEOMETRY 


[§44 


We  have,  then,  the  following  reciprocal  definitions. 


//  two  projectively  related 
sheaves  of  rays  are  coplanar,  but 
are  neither  concentric  nor  in  per- 
spective position,  they  generate  a 
point-row  of  second  order,  or 
curve,  which  has  not  more  than 
two  points  in  common  with  any 
straight  line. 


If  two  projectively  related 
point-rows  are  coplanar,  but  are 
neither  superposed  nor  in  per- 
spective position,  they  generate  a 
sheaf  of  rays  of  second  class,  or 
envelope,  which  has  not  more 
than  two  rays  passing  through 
any  point. 


For,  if  more  than  two  points  lie  on  any  straight  line  (rays  pass 
through  any  point),  the  two  primitive  forms  are  then  in  perspective 
position  (corollary  4). 

It  may  happen  that  a  pair  of  corresponding  elements  does  not 
determine  a  third  element  variable  with  the  pair.  The  primitive 
forms  do  not  then  generate  an  elementary  form.  Thus,  two  pro- 
jectively related  sheaves  of  rays  which  are  neither  coplanar  nor 
concentric  do  not  generate  an  elementary  form,  for  corresponding 
rays  do  not,  in  general,  intersect.  If,  however,  the  sheaves  of 
rays  are  concentric,  corresponding  rays  determine  a  plane  pass- 
ing through  the  common  center,  and  thus  generate  an  elemen- 
tary form.  Again,  projectively  related  sheaves  of  planes  which 
are  not  in  perspective  position  nor  coaxial,  always  generate  an 
elementary  form  since  corresponding  planes  intersect  in  a  line. 
Hence  the  following  definitions: 

Two  projectively  related  sheaves 
of  lines  which  are  concentric,  but 
are  neither  coplanar  nor  in  per- 
spective position,  generate  a 
sheaf  of  planes  of  second  class 
whose  vertex  is  the  common  cen- 
ter. Not  more  than  two  planes 
of  the  sheaf  intersect  in  any  line 
passing  through  the  vertex. 


Two  projectively  related  sheaves 
of  planes  whose  axes  intersect,  but 
which  are  not  in  perspective  posi- 
tion, generate  a  cone  of  second 
order  whose  vertex  is  the  common 
point  of  the  axes.  Not  more 
than  two  rays  of  the  cone  lie  in 
any  plane  passing  through  the 


vertex. 

If  three  rays  of  the  cone  he  in  any  plane  through  the  vertex, 
or  three  planes  of  the  sheaf  intersect  in  any  line  through  the  vertex, 
the  generating  primitive  forms  are  in  perspective  position. 

The  two  elementary  forms  just  defined  are  reciprocals  of  each 


§45]  ELEMENTARY  FORMS  57 

other  in  the  duality  existing  between  a  bundle  of  rays  and  a  bundle 
of  planes  in  which  the  ray  and  the  plane  are  reciprocal  elements. 
Finally,  we  have  the  following  definitions: 


Two  projedively  related  point- 
rows,  which  do  not  lie  in  the 
same  plane,  generate  a  regulus 
no  two  of  whose  rays  can  lie  in 
any  plane. 


Two  protectively  related  sheaves 
of  planes,  whose  axes  do  not 
intersect,  generate  a  regulus  no 
two  of  whose  rays  can  meet  in 
any  point. 


The  reciprocal  of  a  regulus  in  space  is  again  a  regulus. 

If,  from  each  point-row  as  an  axis,  we  project  the  other  point- 
row,  we  obtain  two  projectively  related  sheaves  of  planes  whose 
axes  do  not  intersect.  Corresponding  planes  in  these  two  sheaves 
intersect  in  rays  of  the  regulus.  Reciprocally,  if  we  cut  each  of 
two  generating  sheaves  of  planes  by  the  axis  of  the  other  sheaf, 
we  obtain  two  projectively  related  point-rows  which  generate 
exactly  the  same  regulus. 

If  any  two  rays  of  a  regulus  lie  in  one  plane,  the  point-rows  which 
generate  the  regulus  are  coplanar;  and  the  regulus  itself  becomes 
a  sheaf  of  rays  of  the  second  class. 

That  the  five  elementary  forms  defined  above  are  the  only 
elementary  forms  generated  by  two  projectively  related  primitive 
forms  of  the  first  kind,  will  appear  in  the  sequel. 

The  words  "second  order"  used  in  the  above  definitions  signify 
that  not  more  than  two  points  (rays)  can  lie  on  a  line  (in  a  plane) . 
Reciprocally,  the  words  "second  class"  mean  that  not  more  than 
two  rays  (planes)  can  pass  through  a  point  (ray).  From  this 
point  of  view,  the  primitive  forms  of  the  first  kind  are  of  first  order 
(class). 

45.  Fundamental  Properties  of  the  Curve  and  the  Envelope. — 
Before  discussing  the  relations  existing  between  the  five  elementary 
forms,  we  shall  develop  some  fundamental  properties  of  the 
curve,  or  point-row  of  second  order,  and  its  reciprocal  in  the 
plane,  the  envelope,  or  sheaf  of  rays  of  the  second  class. 

1.  A  point-row  of  second  order  \       A  sheaf  of  rays  of  second  class 


passes  through  the  centers  of  the 
two  projectively  related  sheaves  of 
rays  which  generate  it. 


contains  the  supports  of  the  two 
projectively  related  point-rows, 
which  generate  it. 


58 


PROJECTIVE  GEOMETRY 


[§45 


For  the  line  joining  the  centers  of  the  sheaves,  considered  as 
a  ray  of  one  of  the  sheaves,  must  intersect  its  corresponding  ray 
at  the  center  of  the  other  sheaf.  Similarly,  the  point  common 
to  the  two  point-rows,  considered  as  belonging  to  one  of  the 
point-rows,  is  joined  to  its  corresponding  point  by  the  other 
point-row. 


2.  The  ray  corresponding  to  the 
common  line,  considered  as  a  ray 
of  one  of  the  sheaves,  is  tangent 
to  the  point-row  of  second  order  at 
the  center  of  the  other  sheaf. 


The  point  corresponding  to  the 
common  point,  considered  as  a 
point  of  one  of  the  point-rows, 
is  the  point  of  contact  on  the 
support  of  the  other  point-row. 


In  the  statement  on  the  left,  as  a  variable  ray  p  describes  the 

sheaf  S  (Fig.  36),  its  corresponding 
ray  pi  describes  the  sheaf  Si.  The 
intersection  P  of  p  and  p\  describes 
the  curve  passing  through  S  and  S\ 
(property  1).  AsP  approaches  Si, 
p  approaches  the  position  n  (or  SSi) 
and  the  secant  SiP  (or  p\]  ap- 
proaches a  limiting  position  n\,  cor- 
responding to  n,  which  is  defined  as 
the  tangent  at  Si.  Again,  as  P 

approaches  S,  p\  approaches  the  position  of  the  common  ray  Si  S, 
and  p  approaches  a  limiting  position  m,  defined  as  the  tangent  at  S. 
In  the  statement  on  the  right,  as  a  variable  point  P  describes 
the  point-row  u  (Fig.  37),  its  corresponding  point  PI  describes 
the  point-row  UL  The  line  joining  P  and  PI  describes  the  sheaf 
of  rays  of  second  class,  or  envelope,  which  contains  the  rays  u 
and  MI.  As  P  approaches  the  common  point  M  of  M  and  MI, 
PI  approaches  the  point  MI  corresponding  to  M.  MI  is  called 
the  point  of  contact  on  MI.  Similarly,  as  PI  approaches  the  com- 
mon point  of  M  and  HI  (which  we  now  call  N\  as  belonging  to  the 
point-row  MI),  P  approaches  the  corresponding  point  N  on  M. 
N  is  the  point  of  contact  on  M. 

A  point-row  of  second  order  has  a  tangent  at  each  one  of  its 
points  and  a  sheaf  of  rays  of  second  class  has  a  point  of  contact  on 
each  one  of  its  rays.  Methods  for  constructing  the  tangent  at 


§45] 


ELEMENTARY  FORMS 


59 


any  point  of  a  curve  and  the  point  of  contact  on  any  ray  of  an 
envelope  will  be  developed  later. 


3.  A  point-row  of  second  order 
is  projected  from  any  two  of  its 
points  in  projectively  related 
sheaves  of  rays  in  which  corre- 
sponding rays  intersect  on  the 
point-row. 


A  sheaf  of  rays  of  second  class 
is  cut  by  any  two  of  its  rays  in 
projectively  related  point-rows 
in  which  corresponding  points 
are  joined  by  rays  of  the  sheaf. 


Suppose  the  point-row  of  second  order  is  generated  by  the  two 
projectively  related  sheaves    of  rays  S 
and  Si  (Fig.  38) .     Choose  any  four  points 
on  the  point-row,  as  A,B,C,D,  then 

S(ABCD)^Sl(ABCD) 

by  definition.  Cut  the  sheaves  S  and  Si 
by  the  lines  DC  and  DA  respectively,  and 
thus  obtain  two  projectively  related  point- 
rows  along  these  lines  in  which  the  ranges 
DCTR  and  DMKA  correspond  point  to 
point.  These  point-rows  are  in  perspective  position,  since  their 
common  point  D  corresponds  to  itself  (corollary  3).  Therefore 
the  lines  MC,  KT,  and  RA  are  concurrent  in  $2..  If  the  point 
D  is  allowed  to  describe  the  curve  while  the  points  S,  Si,  A, 
B,  C  remain  fixed,  the  lines  DC  and  DA  will  describe  sheaves  of 


FIG.  38. 


60 


PROJECTIVE  GEOMETRY 


I  §45 


rays  about  C  and  A  respectively.  At  the  same  time,  the  points 
T  and  K  will  describe  point-rows  along  the  fixed  lines  SB  and  S\B 
respectively.  But  these  point-rows  are  in  perspective  position, 
since  they  are  sections  of  the  sheaf  whose  center  is  the  fixed  point 
Sz.  Hence,  we  have  the  chain  of  perspectivity 

C(BSiDS)  ABLTS^BSiKQ^A  (BSiDS)  . 
Consequently 


and  corresponding  rays  meet  in  points  of  the  curve.  Thus  the 
curve  is  also  generated  by  the  sheaves  whose  centers  are  C  and  A  . 
But  C  and  A  are  any  two  points  of  the  curve. 

For  the  reciprocal  statement,  let  the  sheaf  of  rays  of  second 
class  be  generated  by  the  two  protectively  related  point-rows  u 
and  HI  in  which  ABCD  and  AiBiCiDi  are  corresponding  ranges, 
so  that  AAi,  BBi,  CCi,  and  DD\  are  any  four  rays  of  the  sheaf  of 
second  class  (Fig.  39).  Let  AA\  intersect  BBi  and  DD\  in  the 
points  Si  and  S  respectively.  Project  u  and  u\  from  S  and  Si 
respectively,  and  thus  obtain  two  sheaves  of  rays  in  perspective 
position,  since  their  common  ray  corresponds  to  itself.  These 
sheaves  are  then  projectors  of  the  point-row  u^.  If  the  ray  AA\  is 
allowed  to  describe  the  sheaf  of  second  class,  the  points  BCD  and 
BiCiDi  remaining  fixed,  the  points  Si  and  S  will  describe  point- 
rows  along  the  fixed  lines  BBi  and  DDi  in  perspective  position 
with  the  sheaves  of  rays  described  by  SiCi  and  SC  respectively. 
Hence, 


DiSDl 

Consequently 


-AC(DiRBT 


DiSDL 


and  corresponding  points  are  joined  by   rays  of  the  sheaf  of 
second  class.     But  BBi  and  DDi  are  any  two  rays  of  this  sheaf. 
As  an  immediate  consequence  of  the  property  stated  in  (3), 
we  have  the  following: 


4.  A  point-row  of  second 
order  is  completely  determined 
by  any  five  of  its  points. 


A  sheaf  of  rays  of  second 
class  is  completely  determined 
by  any  five  of  its  rays. 


§45] 


ELEMENTARY  FORMS 


61 


For,  if  ABODE  are  any  five  points  of  a  point-row  of  second 
order,  we  can  choose  any  two  of  them  as  centers  of  projectively 
related  sheaves  of  rays  and  these  will  then  generate  the  curve. 
If  A  and  B  are  the  centers  chosen,  then  the  sheaves  are  A  (CDE) 


Si 


|M2 


rs 
FIG.  39. 

and  B(CDE}.     The  three  pairs  of  corresponding  rays  determine 
the  projectivity  (Art.  43,  theorem  VIII). 

Reciprocally,  if  a  b  c  d  e  are  five  rays  of  a  sheaf  of  rays  of  second 
class,  a  projectivity  is  determined  between  any  two  of  them  by 
the  remaining  three.  As  many  rays  of  the  sheaf  of  second  class 
can  then  be  constructed  as  may  be  desired. 


62 


PROJECTIVE  GEOMETRY 


[§46 


46.  Construction  of  Curves  and  Envelopes. — By  means  of 
the  properties  stated  in  Art.  45  and  with  the  aid  of  theorem  VIII, 
we  are  able  to  solve  the  following  construction  problems,  viz.: 


To  construct  a  curve  of  second 
order  having  given  (a)  five  points, 
(b)  four  points  and  the  tangent 
at  one  of  them,  (c)  three  points 
and  the  tangents  at  two  of  them. 


To  construct  an  envelope  of 
second  class  having  given  (a) 
five  rays,  (b)  four  rays  and  the 
point  of  contact  on  one  of  them, 
(c)  three  rays  and  the  points  of 
contact  on  two  of  them. 

For,  in  each  case,  the  given  elements  are  just  sufficient  to  deter- 
mine a  projectivity  between  two  primitive  forms  and  these  will 


FIG.  40. 


FIG.  41. 


then  generate  the  elementary  form  required.  Thus,  in  case  (c) 
on  the  left,  let  S,  Si  and  B  be  the  three  given  points  and  a  and  c\, 
the  tangents  at  S  and  Si  respectively  (Fig.  40).  The  three  pairs 
of  corresponding  rays,  aa\,  bbi,  cci,  completely  determine  a  pro- 
jectivity between  the  sheaves  of  rays  S  and  Si  (theorem  VIII), 
and  these  generate  a  curve  of  second  order  which  passes  through 
S,  Si,  and  B  and  is  tangent  to  a  and  Ci  at  S  and  Si  respectively,  by 
virtue  of  the  properties  stated  in  (1)  and  (2),  Art.  45.  As  many 
points  of  the  curve  can  be  constructed  as  may  be  desired. 

Again,  in  case  (b)  on  the  right,  if  AC,  A\C\,  BB\,  and  CC\  are 
the  four  given  rays  and  A  is  the  given  point  of  contact  on  AC 
(Fig.  41),  then  the  pairs  of  points  A,  AI,  B,  B\,  and  C,  Ci  determine 


§46]  ELEMENTARY  FORMS  63 

a  projectivity  between  the  rays  AC  and  A\C\.  These  projectively 
related  point-rows  generate  a  sheaf  of  rays  of  second  class  con- 
taining the  four  given  rays  and  having  A  as  the  point  of  contact 
on  AC.  As  many  rays  of  this  sheaf  can  then  be  constructed  as 
may  be  desired. 

Special  cases  may  arise.  Thus,  if  three  of  the  five  given  points 
in  (a)  on  the  left  are  collinear,  the  sheaves  formed  by  projecting 
any  three  of  the  points  from  the  remaining  two  are  in  perspective 
position  and  thus  generate  a  point-row  of  the  first  order  (Art.  )44. 

Reciprocally,  if  three  of  five  given  rays  are  concurrent,  the  point- 
rows  cut  by  any  three  of  the  rays  upon  the  remaining  two  are  in 
perspective  position  and  thus  generate  a  sheaf  of  rays  of  the  first 
class. 

The  construction  in  each  case  is  unique  (theorem  VIII)  and 
hence  we  conclude  that: 


Two  curves  of  second  order 
coincide  if  they  have  in  common 
five  points,  or  four  points  and  the 
tangent  at  one  of  them,  or  three 
points  and  the  tangents  at  two  of 
them. 


Two  envelopes  of  second  class 
coincide  if  they  have  in  common 
five  rays,  or  four  rays  and  the 
point  of  contact  on  one  of  them, 
or  three  rays  and  the  points  of 
contact  on  two  of  them. 


Exercises 

1.  Construct  the  point-row  of  second  order  determined  by  five 
given  points  two  of  which  are  ideal. 

2.  Construct  the  sheaf  of  rays  of  second  class  determined  by  five 
given  rays  one  of  which  is  the  ideal  line. 

3.  Construct  the  point-row  of  second  order  determined  by  three 
points,  two  of  which  are  ideal  points,  and  the  tangents  at  the  ideal 
points. 

4.  The  centers  of  two  sheaves  of  rays  lie  on  a  circle  and  correspond- 
ing rays  meet  in  points  of  the  circle.     Can  the  sheaves  be  super- 
posed so  as  to  coincide  throughout?     Are  the  sheaves  projectively 
related?     Is  the  circle  a  curve  of  second  order? 

5.  A  variable  tangent  to  a  circle  cuts  out  point-rows  on  two  fixed 
tangents  to  the  same  circle.     What  angle  does  the  segment  of  the 
variable  tangent  contained  between  the  fixed  tangents  subtend  at  the 
center  of  the  circle?     Is  this  angle  constant?     Are  the  point-rows  cut 
out  on  the  fixed  tangents  projectively  related?     Is  the  system  of 
tangents  to  a  circle  a  sheaf  of  rays  of  second  class? 


64  PROJECTIVE  GEOMETRY  [§47 

6.  A  variable  triangle  ASA  i  moves  in  its  plane  so  that  the  vertices 
A  and  A\  continually  lie  upon  straight  lines  u  and  u\  respectively, 
while  the  sides  SA  and  SAi  rotate  about  the  fixed  point  S,  the  angle 
S   remaining   of   constant   magnitude.     Show   that   the   base    AAi 
describes  a  sheaf  of  rays  of  second  class  to  which  u  and  Ui  belong. 

7.  Given  a  plane  quadrangle  KLMN.     What  is  the  locus  of  a  point 
S  such  that  the  pencil  S(KLMN)  is  always  harmonic? 

8.  If  a,  b,  c,  d,  e,  are  five  rays  of  an  envelope  of  second  class,  con- 
struct the  points  of  contact  on  b  and  c. 

9.  If  A,  B,   C,  D,  E  are  five  points  of  a  curve  of  second  order, 
construct  the  tangents  at  A  and  B. 

10.  In  the  preceding  exercise,   A   and  B  are  ideal  points.     The 
tangents  at  A  and  B  are  asymptotes.     Construct  the  asymptotes. 

11.  If  the  point  of  contact  on  any  ray  of  an  envelope  of  second  class 
is  infinitely  distant,  that  ray  is  an  asymptote.     Construct  an  envelope 
of  second  class  having  given  two  asymptotes  and  one  other  ray. 

12.  A  triangle  moves  so  that  its  vertices  continually  lie  upon  three 
fixed  lines  while  two  of  its  sides  rotate  about  two  fixed  points.     Show 
that  the  third  side  will,  in  general,  describe  an  envelope  of  second  class. 
Under  what  circumstances  will  the  envelope  be  of  first  class? 

13.  What  is  the  reciprocal  of  the  preceding  exercise  in  the  plane? 

14.  Two  angles  P  and  Q  of  fixed  magnitude  and  lying  in  the  same 
plane  rotate  about  P  and  Q  while  the  intersection  of  two  sides  trav- 
erses a  fixed  line.     Show  that  the  intersection  of  the  other  two  sides 
will,  in  general,  describe  a  curve  of  second  order  (Newton's  method  of 
generating  conies). 

47.  Relations  Existing  among  the  Elementary  Forms. — The 
elementary  forms  are  related  one  to  another  by  means  of  projec- 
tion and  section.  Thus,  the  section  of  a  cone,  by  a  plane  not 
passing  through  its  vertex,  is  a  curve  of  second  order.  For  the 
plane  cuts  the  two  sheaves  of  planes  which  generate  the  cone 
in  two  sheaves  of  rays  which  generate  the  curve.  Corresponding 
planes  are  cut  in  corresponding  rays,  the  latter  intersect  upon  that 
ray  of  the  cone  which  is  determined  by  the  former. 

Similarly,  the  section  of  a  sheaf  of  planes  of  second  class,  by  a 
plane  not  passing  through  its  vertex,  is  an  envelope  of  second 
class.  For  the  plane  cuts  the  two  sheaves  of  rays  which  generate 
the  sheaf  of  planes  in  two  point-rows  which  generate  the  envelope. 
Corresponding  rays  are  cut  in  corresponding  points,  and  the  latter 
are  joined  by  the  trace  of  the  plane  determined  by  the  former. 


ELEMENTARY  FORMS 


65 


Again,  if  we  project  a  curve  of  second  order  from  a  point  not 
lying  in  its  plane,  we  obtain  a  cone  of  second  order.  For  the  pro- 
jectors of  the  two  sheaves  of  rays  which  generate  the  curve  are  two 
sheaves  of  planes  which  generate  the  cone. 

In  the  same  way,  the  projector  of  an  envelope  of  second  class 
is  a  sheaf  of  planes  of  second  class,  since  the  projectors  of  the 
point-rows  that  generate  the  envelope  are  sheaves  of  rays  that 
generate  the  sheaf  of  planes. 

For  the  regulus,  we  have  the  following  dual  statements: 

The  section  of  a  regulus,  by  a  The  -projector  of  a  regulus, 
'plane  not  passing  through  any  from  a  point  not  lying  on  any 
ray,  is  a  curve  of  second  order,  ray,  is  a  sheaf  of  planes  of 

second  class.  For  the  projectors 
of  the  generating  point-rows  are 
sheaves  of  rays  which  generate 
the  sheaf  of  planes. 
That  the  elementary  forms  are  also  related  one  to  another  by 
means  of  the  principle  of  duality  has  been  exemplified  in  the 
definitions  (Art.  44)  and  in  the  properties  developed  in  Art.  45. 
There  are,  however,  other  reciprocal  relations  of  importance. 
Thus,  the  reciprocal  of  a  curve  of  second  order  in  space  is  a  sheaf 
of  planes  of  second  class,  since: 


For  the  plane  cuts  the  generating 
sheaves  of  planes  in  sheaves  of 
rays  which  generate  the  curve. 


The  curve  is  composed  of  points 
lying  in  a  plane,  no  three  of  which 
can  lie  on  any  straight  line. 


The  sheaf  is  composed  of 
planes  passing  through  a  point, 
no  three  of  which  can  meet  in  any 
straight  line. 

Again,  an  envelope  of  second  class  is  the  space-dual  of  a  cone 
of  second  order,  since: 


The  envelope  consists  of  rays 
lying  in  a  plane,  no  three  of  which 
can  meet  in  any  point. 


The  cone  consists  of  rays  pass- 
ing through  a  point,  no  three  of 
which  can  lie  in  any  plane. 


Exercises 

1.  By  means  of  projection,  derive  the  properties  of  a  cone  and  of  a 
sheaf  of  planes  of  second  class  corresponding  to  the  properties  of  a 
curve  and  of  an  envelope  stated  in  Art.  45. 

2.  Derive  the  same  properties  by  means  of  the  principle  of  duality. 

5 


66  PROJECTIVE  GEOMETRY  [§48 

48.  Totality  of  Elementary  Forms. — That  the  elementary  forms 
defined  in  Art.  44  are  the  only  ones  generated  by  two  projectively 
related  primitive  forms  of  the  first  kind  follows  upon  comparing 
the  primitive  forms  of  the  first  kind,  one  with  another,  in  all 
possible  ways.  We  have  already  considered  the  cases  where  the 
two  primitive  forms  consist  of  like  elements.  The  only  other 
cases  in  which  two  projectively  related  primitive  forms  of  the  first 
kind  can  generate  an  elementary  form  are: 

1.  A  sheaf  of  rays  projectively  related  to  a  sheaf  of  planes. 

2.  A  sheaf  of  rays  projectively  related  to  a  point-row. 

In  the  first  case,  if  a  is  the  plane  containing  the  given  sheaf  of 
rays,  the  elementary  form  is  likewise  generated  by  the  given  sheaf 
of  rays  and  the  section  of  the  sheaf  of  planes  by  a,  and  is,  therefore, 
a  curve  of  second  order. 

In  the  second  case,  if  S  is  the  center  of  the  given  sheaf  of  rays, 
the  elementary  form  is  also  generated  by  the  given  sheaf  of  rays 
and  the  projector  of  the  point-row  from  S,  and  is  consequently  a 
sheaf  of  planes  of  second  class. 

We  conclude,  therefore,  that  there  are  only  five  elementary 
forms  generated  by  projectively  related  primitive  forms  of  the 
first  kind,  namely,  those  defined  in  Art.  44. 

49.  Classification  of  Curves  of  Second  Order. — Since  a  curve 
of  second  order  cannot  meet  any  straight  line  in  more  than  two 
points  (Art.  44),  it  cannot  have  more  than  two  points  in  common 
with  the  ideal  line  in  its  plane.     If  the  curve  does  not  meet  the 
ideal  line,  it  is  called  an  ellipse :  if  it  has  one  point  in  common  with 
the  ideal  line;  that  is,  if  it  is  tangent  to  the  ideal  line,  it  is  called 
a  parabola :  and  if  it  meets  the  ideal  line  in  two  distinct  points,  it 
is  called  a  hyperbola. 

The  ellipse  lies  wholly  in  the  finite  part  of  the  plane,  the  parabola 
stretches  out  indefinitely  in  one  direction  toward  the  infinitely 
distant  part  of  the  plane,  while  the  hyperbola  crosses  the  ideal 
line  and  appears  on  both  sides  of  it. 

The  tangents  to  the  hyperbola  at  the  ideal  points  are  called  the 
asymptotes.  If  the  asymptotes  are  perpendicular  to  each  other, 
the  hyperbola  is  equilateral. 

50.  The  Conic  Sections.- — That  the  ellipse,  the  parabola,  and 
the  hyperbola  are  sections  of  a  cone  of  second  order  whose  vertex 


§50]  ELEMENTARY  FORMS  67 

is  not  an  ideal  point  may  be  seen  as  follows:  A  plane  a  passing 
through  the  vertex  must  meet  the  cone  in  this  point  alone,  or 
touch  the  cone  along  one  of  its  rays  s,  or  cut  the  cone  in  two  of  its 
rays  p  and  q.  Any  plane  parallel  to  a  must,  then,  cut  all  the  rays 
of  the  cone  in  actual  points,  or  cut  all  the  rays  except  s  in  actual 
points,  or  cut  all  the  rays  except  p  and  q  in  actual  points.  In 
the  first  case,  the  section  of  the  cone  is  an  ellipse;  in  the  sec- 
ond, a  parabola;  and  in  the  third,  a  hyperbola.  Hence,  in  the 
future,  we  can  call  the  curves  of  second  order  conic  sections,  or 
more  briefly  conies. 

A  further  development  of  the  properties  of  the  conies  and  their 
reciprocals,  the  envelopes  of  second  class,  requires  the  aid  of  two 
important  and  historic  theorems  which  are  reciprocals  of  each 
other.  These  theorems  will  be  the  subject  of  the  next  chapter. 

Exercises 

1.  If  the  axes  of  two  protectively  related  sheaves  of  planes   are 
parallel  to  each  other,  what  elementary  form  is  generated  by  them? 

2.  If  from  a  given  point  perpendicular  lines  are  dropped  upon  the 
planes  of  a  sheaf  of  first  class,  what  is  the  locus  of  the  feet  of  these 
perpendiculars? 

3.  If  from  a  given  point  perpendicular  lines  are  dropped  upon  the 
planes  of  a  sheaf  of  second  class,  show  that  these  lines  lie  upon  a  cone 
of  second  order. 

4.  If  from  a  point  outside  the  plane  of  a  sheaf  of  rays  of  first  class, 
perpendicular  planes  are  drawn  to  the  rays  of  the  sheaf,  these  planes 
will  form  a  sheaf  of  planes  of  first  class.     What  line  is  the  axis  of  the 
sheaf? 

5.  Construct  a  hyperbola  having  given  the  two  asymptotes  and 
one  point. 

Suggestion. — Take  the  ideal  points  on  the  given  asymptotes  as 
the  centers  of  two  projectively  related  sheaves  of  rays. 

6.  If  two  projectively  related  sheaves  of  rays  are  coplanar,  but  not 
concentric,  and  are  oppositely  projective  (Art.  40),  show  that  they 
must  generate  a  hyperbola.     If  they  are  directly  projective,  would 
they  necessarily  generate  an  ellipse?     Why? 


CHAPTER  VII 


THE  PASCAL  THEOREM  AND  THE  BRIANCHON  THEOREM 

51.  Six  Elements  of  a  Conic  or  of  an  Envelope. — Five  points  in 
a  plane,  no  three  of  which  lie  on  any  line,  completely  determine  a 
conic  (Art.  45).  Reciprocally,  five  rays  in  a  plane,  no  three  of 
which  pass  through  any  point,  completely  determine  an  envelope. 
New  properties  of  the  conies  and  of  their  reciprocals,  the  envelopes, 
are  derived  from  the  geometrical  condition  that  must  be  satisfied 
in  order  that  a  sixth  element  may  belong  to  the  conic,  or  to  the 
envelope,  determined  by  any  given  five  elements.  The  condition 
for  the  conic  is  known  as  the  Pascal  theorem,  and  the  correspond- 
ing condition  for  the  envelope  is  called  the  Brianchon  theorem, 
from  their  respective  discoverers. 


The  Brianchon  Theorem.— 
The  opposite  vertices  of  any 
simple  hexagon  whose  sides  are 
rays  of  an  envelope  of  second 
class  are  joined  by  three  con- 
current lines. 


Theorems  IX. — The  Pascal 
Theorem. — The  opposite  sides  of 
any  simple  hexagon  whose  vertices 
are  points  of  a  curve  of  second 
order  intersect  in  three  collinear 
points. 

For  the  Pascal  theorem,  let  A,  B,  C,  D,  E,  F  be  the  vertices  of 
the  hexagon  (Fig.  42).  Since  the  conic 
can  be  generated  by  sheaves  of  rays  whose 
centers  are  any  two  of  these  points  (Art. 
45),  we  may  choose  A  and  Cas  the  centers. 

The  sheaves  are  then  A  (BDEF )  and 

C(BDEF ) .     Cut  these  sheaves  by  the 

lines  ED  and  EF  respectively,  and  thus 
obtain   the   point-rows   LDEM —  -  and 
pIG    42  SREF .     These  point-rows  are  in  per- 

spective position  (Art.  42,  corollary  3),  and 

hence  LS,  DR,  and  MF  meet  in  a  point  U.    Thus  the  opposite 
sides  of  the  hexagon  ABCDEF  intersect  on  the  line  LUX, 

68 


§52] 


PASCAL  AND  BRIANCHON  THEOREM 


69 


For  the  Brianchon  theorem,  let  a,  b,  c,  d,  e,  f  be  the  sides  of  the 
hexagon  (Fig.  43).  Since  the  envelope  of  second  class  is  cut  by 
any  two  of  these  lines  in  protectively  related  point-rows  (Art.  45), 
we  may  cut  the  envelope  by  the  lines  a  and  c,  and  thus  obtain  the 

projectively  related  point-rows  ABCD —  -  and  A\B\C\Di . 

Let  the  sides  d  and  /  meet  the  side  e  in  the  points  S  and  Si  respec- 
tively.    Project  ABCD from  S  and  AiBiCiDi from  Si, 

and  we  obtain  two  sheaves  of  rays  in  perspective  position  (corollary^ 
3)  whose  corresponding  rays  meet  on  the  line  B\D.     But  SA  and 


FIG.  43. 

SiAi  are  corresponding  rays.  Therefore  opposite  vertices  of  the 
hexagon  are  joined  by  concurrent  Jines. 

52.  Converse  Theorems. — The  statements  of  the  Pascal  theorem 
and  of  the  Brianchon  theorem  in  the  preceding  article  are  the 
ordinary  ones.  The  proofs  show  that  the  conditions  embodied 
in  the  statements  are  necessary  in  order  that  six  elements  shall 
belong  to  a  curve  or  to  an  envelope.  To  show  that  these  con- 
ditions are  also  sufficient,  we  must  prove  the  converse  of  each 
theorem.  That  is,  for  the  Pascal  theorem,  we  must  show  that: 

//  the  opposite  sides  of  any  simple  hexagon  intersect  in  three 
collinear  points,  the  vertices  are  points  of  a  curve  of  second  order. 


70  PROJECTIVE  GEOMETRY  [§52 

There  is  no  difficulty  in  doing  this.  Thus,  if  A,  B,  C,  D,  E,  F 
are  the  vertices  of  any  simple  hexagon  (Fig.  42),  the  sheaves 

A  (BDF )  and  C(BDF )  generate  a  conic  passing  through 

the  five  points  A,  B,  C,  D,  F  (Art.  45).  If  the  opposite  sides  of 
the  hexagon  meet  in  three  collinear  points,  the  point-rows  cut  upon 

the  lines  ED  and  EF  by  the  sheaves  A  (BDF )  and  C(BDF ) , 

respectively,  are  in  perspective  position,  since  they  are  sections 
of  the  sheaf  whose  center  is  the  point  U  in  which  the  lines  AF  and 
CD  intersect.  Consequently  the  point  E  corresponds  to  itself  in 
the  two  point-rows,  and  the  rays  AE  and  CE  correspond  to  each 
other  in  the  sheaves  which  generate  the  curve.  Hence  the  conic 
passes  through  E,  and  thus  contains  all  six  of  the  vertices  of  the 
hexagon. 

A  special  case  arises  if  the  vertices  B,  D,  F  are  collinear.  In  this 

case,  the  sheaves  A(BDF )  and  C(BDF )  are  in  perspective 

position  (corollary  4)  and  thus  generate  a  point-row  of  first  order 
passing  through  the  points  B,  D,  F.  But  since  the  rays  AE  and 
CE  correspond  to  each  other,  as  above,  they  are  the  projectors  of 

the  same  point  on  the  point-row  BDF ,  and  hence  the  vertices 

A,  C,  E  are  also  collinear  (cf.  Art.  10,  exercise  4,  also  Art.  43,  exer- 
cise 8).  In  this  case,  the  curve  of  second  order  is  said  to  de- 
generate into  two  point-rows  of  first  order,  namely,  the  point- 
rows  DBF and  ACE .  The  figure  consists  of  nine  points 

and  nine  lines  so  situated  that  three  points  lie  on  each  line  and  three 
lines  pass  through  each  point.  This  arrangement  of  points  and 
lines  in  a  plane  is  known  as  a  configuration  of  Pappus  (cf.  configura- 
tion of  Desargues,  Art.  13). 

'  The  converse  of  the  Brianchon  theorem  can  be  proved  in  a  like 
manner.  There  is  also  a  special  case  if  three  sides  of  the  hexagon 
meet  in  a  point.  The  resulting  combination  of  points  and  lines  is 
a  configuration  of  Pappus.  In  this  special  case  the  envelope  of 
second  class  is  said  to  degenerate  into  two  sheaves  of  rays  of  first 
class. 

Exercise 

State  and  prove  the  converse  of  the  Brianchon  theorem.  Also 
state  and  prove  the  special  case  when  three  sides  of  the  hexagon  meet 
in  a  point.  Draw  figures  to  illustrate. 


§53] 


PASCAL  AND  BRIANCHON  THEOREM 


71 


53.  Application  of  Theorems  IX. — The  Pascal  theorem  and  the 
Brianchon  theorem  enable  us  to  solve  the  following  construction 
problems : 

Given  five  points  of  a  curve  of 
second  order;  with  the  aid  of  a 
ruler  to  determine  the  second 
point  of  the  curve  upon  any 
straight  line  drawn  through  one 
of  the  given  points. 


Given  five  rays  of  an  envelop6 
of  second  class;  with  the  aid  of  a 
ruler  to  determine  the  second  ray 
of  the  envelope  passing  through 
any  point  on  one  of  the  given  rays. 


FIG.  44. 


Let  A,  B,  C,  D,  E  be  the  five 
given  points  and  suppose  any 
line,  as  /,  is  drawn  through  one 
of  them,  say  A  (Fig.  44).  Join 
these  points  two  and  two  so  as 
to  form  with  /  five  sides  of  a 


Let  a,  b,  c,  d,  e  be  the  five 
given  rays  and  suppose  any 
point,  as  F,  is  chosen  on  one  of 
them,  say  a  (Fig.  45).  Find 
the  intersection  of  these  lines 
two  and  two  so  as  to  form  with 


72 


PROJECTIVE  GEOMETRY 


[§54 


simple  hexagon.  Let  a,  b,  c,  d 
be  the  lines  so  constructed. 
The  problem  now  is  to  con- 
struct the  sixth  side,  e,  so  that 
opposite  sides  of  the  hexagon 
intersect  in  three  collinear 
points.  If  we  arrange  the 
sides  thus: 

a  &  c 
def, 

then  the  opposite  sides  are 
readily  seen  to  be  a  and  d,  b  and 
e,  c  and  /.  But  the  points  (ad) 
and  (c/)  are  joined  by  the  line 
u.  Hence  (be)  lies  on  u,  and 
(ef),  or  F,  is  the  point  of  the 
curve  sought. 

The  line  u  is  called  a  Pascal 
line. 

By  drawing  a  series  of  lines 
through  A,  we  can  construct  as 
many  points  of  the  curve  as 
may  be  desired. 


F  five  vertices  of  a  simple  hexa- 
gon. Let  A,  B,  C,  D  be  the 
points  so  determined.  The 
problem  now  is  to  construct  the 
sixth  vertex,  E,  so  that  opposite 
vertices  of  the  hexagon  are 
joined  by  three  concurrent  lines. 
If  we  arrange  the  vertices  thus : 
ABC 
DEF, 

then  the  opposite  vertices  are 
readily  seen  to  be  A  and  D,  B 
and  E,  C  and  F.  But  the 
lines  AD  and  CF  intersect  in 
the  point  U.  Hence  BE  passes 
through  U,  and  EF,  or  /,  is  the 
ray  of  the  envelope  sought. 

The    point    U    is    called    a 
Brianchon  point. 

By  choosing  a  series  of  points 
on  a,  we  can  construct  as  many 
rays  of  the  envelope  as  may  be 
•  desired. 


Exercises 

1.  A  conic  is  given  by  five  points  in  a  plane  of  which  two  are  ideal 
points;  construct  the  second  point  of  intersection  with  the  conic  of 
any  line  through  one  of  the  ideal  points. 

2.  A  conic  is  given  by  five  points  of  which  one  is  an  ideal  point; 
construct  the  second  point  of  intersection  of  the  conic  with  the  ideal 
line. 

3.  A  sheaf  of  rays  of  second  class  is  given  by  five  rays  of  which  one 
is  the  ideal  line  in  the  plane;  determine  the  ray  of  the  sheaf  which  passes 
through  any  given  ideal  point.     By  choosing  a  series  of  ideal  points, 
construct  a  number  of  rays  of  the  sheaf. 

54.  Degenerate  Cases  of  Theorems  IX. — If,  in  the  Pascal 
theorem,  two  vertices  of  the  hexagon  approach  coincidence,  the 
side  joining  them  approaches  a  limiting  position  which  is  the 


§55] 


PASCAL  AND  BRIANCHON  THEOREM 


73 


tangent  at  that  point  of  the  conic  through  which  impasses.  The 
hexagon  then  degenerates  into  a  pentagon  with  the  tangent  at  one 
vertex. 

It  is  clear  that  the  hexagon  may  also  degenerate  into  a  quad- 
rangle with  the  tangents  at  two  of  the  vertices,  or  into  a  triangle 
with  the  tangents  at  the  three  vertices. 

In  the  Brianchon  theorem,  two  sides  of  the  hexagon  may  ap- 
proach coincidence;  their  point  of  intersection  then  approaches  a 
limiting  position  which  is  the  point  of  contact  on  that  ray  of  the 
envelope  which  passes  through  it.  The  hexagon  then  degener- 
ates into  a  pentagon  with  the  point  of  contact  on  one  of  its  sides. 

The  hexagon  may  also  degenerate  into  a  quadrangle  with  the 
points  of  contact  on  two  of  its  sides,  or  into  a  triangle  with  the 
points  of  contact  on  the  three  sides. 

These  degenerate  cases  of  theorems  IX  and  some  applications 
of  them  are  taken  up  in  the  following  articles. 

55.  The  Pentagon  Theorem  and  Its  Dual. — 


//  the  vertices  of  any  simple 
pentagon  are  points  of  a  given 
ciirve  of  second  order,  then  two 
pairs  of  non-adjacent  sides  inter- 
sect in  points  collinear  with  the 
intersection  of  the  remaining  side 
and  the  tangent  at  the  opposite 
vertex. 

For,  if  a,  b,  c,  d,  e  are  the 
sides  of  the  pentagon  (Fig.  46) 
and  /  is  the  tangent  at  one  ver- 
tex, say  (be),  then  the  sides  of 
the  degenerate  hexagon  are  in 
order  a,  b,  f,  c,  d,  e.  The  three 
points  (ac),  (bd),  (ef)  are  collin- 
ear by  the  Pascal  theorem;  a 
and  c,  b  and  d  are  pairs  of  non- 
adjacent  sides  and  the  vertex 
(be)  is  opposite  the  side  e. 


If  the  sides  of  any  simple 
pentagon  are  rays  of  a  given 
envelope  of  second  class,  then  two 
pairs  of  non-adjacent  vertices 
are  joined  by  lines  concurrent 
with  the  line  joining  the  re- 
maining vertex  to  the  point  of 
contact  on  the  opposite  side. 

For,  if  A,  B,  C,  D,  E  are  the 
vertices  of  the  pentagon  (Fig. 
47)  and  F  is  the  point  of  contact 
on  one  side,  say  BC,  then  the 
vertices  of  the  degenerate  hexa- 
gon are  in  order  A,  B,  F,  C, 
D,E.  The  three  lines  A C,BD, 
EF  are  concurrent  by  the 
Brianchon  theorem;  A  and  C,  B 
and  D  are  pairs  of  non -adja- 
cent vertices  and  the  side  BC 
is  opposite  the  vertex  E. 


74 


PROJECTIVE  GEOMETRY 


[§56 


FIG.  46. 


FIG.  47. 


56.  Application  of  the  Pentagon  Theorem  and  Its  Dual. — 


A  curve  of  second  order  is  given 
by  five  points;  with  the  aid  of  a 
ruler,  construct  the  tangent  to  the 
curve  at  each  of  the  given  points. 
The  five  given  points  are 
vertices  of  a  simple  pentagon. 
Consider  in  turn  each  degene- 
rate hexagon  whose  sides  are 
the  five  sides  of  the  pentagon 
and  the  tangent  at  one  vertex. 


An  envelope  of  second  class  is 
given  by  five  rays;  with  the  aid  of 
a  ruler,  construct  the  point  of 
contact  on  each  of  the  given  rays. 

The  five  given  rays  are. sides 
of  a  simple  pentagon.  Con- 
sider in  turn  each  degenerate 
hexagon  whose  vertices  are  the 
five  vertices  of  the  pentagon 
and  the  point  of  contact  on  one 
side. 


Exercises 


1.  Given  five  points  of  a  curve  of  second  order  of  which  two  are 
ideal  points;  construct  the  tangents  at  the  ideal  points. 


§56] 


PASCAL  AND  BRIANCHON  THEOREM 


75 


2.  Given  five  rays  of  an  envelope  of  second  class  of  which  one  is 
the  ideal  line;  construct  the  points  of  contact  on  each  of  the  given 
rays. 


FIG.  48. 

3.  Given  three  points  of  a  conic  and  the  tangents  at  two  of  them. 
Through  one  of  the  points  a  straight  line  is  drawn;  construct  the 
second  point  of  intersection  of  this  line  with  the  curve. 

4.  Given  four  rays  of  an  envelope  of  second  class  and  the  point  of 
contact  on  one  of  them.     A  point  is  chosen  on  one  of  the  given  rays; 


76 


PROJECTIVE  GEOMETRY 


construct  the  second  ray  of  the  envelope  which  passes  through  this 
point. 

57.  The  Quadrangle  Theorem  and  Its  Dual. — 


//  the  vertices  of  any  simple 
quadrangle  are  points  of  a  given 
curve  of  second  order,  then  pairs 
of  opposite  sides  of  the  quadran- 
gle intersect  in  points  collinear 
with  the  points  of  intersection  of 
tangents  at  opposite  vertices. 

The  sides  of  the  quadrangle 
together  with  the  tangents  at  a 
pair  of  opposite  verticles  form 
a  degenerate  hexagon.  Thus 
(Fig.  48),  akdcmb  and  adncbl 
are  degenerate  hexagons  of 
of  which  XZ  is  the  common 
Pascal  line. 


//  the  sides  of  any  simple 
quadrangle  are  rays  of  a  given 
envelope  of  second  class,  then 
the  diagonals  of  the  quadrangle 
are  concurrent  with  the  lines 
joining  the  points  of  contact  on 
opposite  sides. 

The  vertices  of  the  quad- 
rangle together  with  the  points 
of  contact  on  a  pair  of  opposite 
sides  form  a  degenerate  hexa- 
gon. Thus  (Fig.  48),  AKDCMB 
and  ADNCBL  are  degenerate 
hexagons  of  which  Y  is  the 
common  Brianchon  point. 


58.  Application  of  the  Quadrangle  Theorem  and  Its  Dual. 


The  tangents  to  a  curve  of 
second  order  form  a  sheaf  of  rays 
of  second  class. 


The  points  of  contact  on  the 
rays  of  an  envelope  of  second  class 
form  a  curve  of  second  order. 


If,  in  Fig.  48,  we  allow  the  point  K  to  move  along  the  curve  while 
the  points  L,  M,  and  N  remain  fixed,  the  lines  DB  and  AC  will 
describe  sheaves  of  rays  about  the  fixed  points  B  and  C  respectively. 
But  these  sheaves  are  in  perspective  position,  since  correspond- 
ing rays  intersect  on  the  fixed  line  LN ,  and  they  cut  out  pro- 
jectively  related  point-rows  along  the  fixed  tangents  I  and  n 
respectively.  Corresponding  points  on  these  point-rows  are  joined 
by  the  various  positions  of  the  tangent  k.  Thus  k  describes  an 
envelope  of  second  class. 

On  the  other  hand,  if  we  allow  k  to  describe  an  envelope  of 
second  class  while  I,  m,  and  n  remain  fixed,  the  sheaves  described 
about  the  fixed  points  B  and  C  by  the  varying  lines  DB  and  AC 
will  cut  out  point -rows  along  the  fixed  lines  LM  and  MN  respec- 


§60]  PASCAL  AND  BRIANCHON  THEOREM  77 

tively.  But  these  point-rows  are  in  perspective  position,  since  cor- 
responding points,  as  X  and  Z,  are  joined  by  rays  through  the  fixed 
point  S.  Project  the  point-rows  thus  determined  on  LM  and  MN 
from  the  fixed  points  N  and  L,  respectively,  and  we  obtain  pro- 
jectively  related  sheaves  whose  corresponding  rays  intersect  in  the 
various  positions  of  the  point  of  contact  K.  Therefore,  K  de- 
scribes a  curve  of  second  order. 

The  theorems  just  proved  show  the  relation  between  a  curve  of 
second  order  and  an  envelope  of  second  class.  From  it  we  see 
that  not  more  than  two  tangents  to  a  conic  can  pass  through  any 
point  in  the  plane. 

A  point  through  which  pass  two  tangents  to  a  conic  is  said  to  be 
outside  the  curve.  If  no  tangents  pass  through  the  point,  it  is 
inside  the  curve.  If  one  tangent  passes  through  the  point,  it  is 
on  the  curve  and  is  the  point  of  contact  of  the  tangent  which  passes 
through  it. 

59.  The   Principle  of   Continuity.— The  proofs  given  for  the 
special  cases  in  the  preceding  articles  depend  upon  the  so-called 
principle  of  continuity.     This  principle  was  formulated  by  Ponce- 
let  (1822)  and  asserts  that  properties  of  a  geometrical  figure  which 
hold  when  the  figure  varies  according  to  definite  laws  will  also  hold 
when  the  figure  assumes  a  limiting  position.     Thus,  the  Pascal 
theorem  holds  as  long  as  the  six  vertices  of  the  hexagon  are  on  the 
curve,  and  the  principle  of  continuity  asserts  that  the  theorem 
holds  also  when  the  hexagon  assumes  a  limiting  position;  that  is, 
one  or  another  of  the  degenerate  cases  stated  in  Art.  54.     But  the 
principle  of  continuity  rests  upon  intuitive  grounds  rather  than 
upon  logical  rigor.     It  is  not  necessary,  however,  to  enter  upon  a 
discussion  of  its  validity,  since  any  one  of  the  special  cases  is  easily 
proved  quite  independently  of  the  Pascal,  or  the  Brianchon, 
theorem. 

On  account  of  its  importance,  we  shall  demonstrate  the  quad- 
rangle theorem  and  its  dual  without  reference  to  the  Pascal 
theorem  or  the  Brianchon  theorem. 

60.  Second  Proof  of  the  Quadrangle  Theorem  and  Its  Dual. — 
Let    KLMN    be    any    four         Let  klmn  be  any  four  rays 

points   of   a   curve   of   second  j  of  an  envelope  of  second  class 


78 


PROJECTIVE  GEOMETRY 


[§61 


order  and  k  and  m  the  tangents 
at  K  and  M  respectively  (Fig. 
48).  The  curve  is  then  gen- 
erated by  the  two  projec- 
tively  related  sheaves  of  rays 
K(FNML)  and  M(KNFL) 
(Art.  45).  When  we  cut  these 
sheaves  by  the  lines  ML  and 
KL,  respectively,  we  obtain  the 
two  point-rows  TZML  and 
KXT'L  which  are  in  perspec- 
tive position,  since  they  have 
the  point  L  as  a  self-corre- 
sponding point.  The  point  F 
is  the  center  of  perspectivity. 
Hence,  the  intersection  of  the 
tangents  at  the  opposite  ver- 
tices K,  M  lies  on  the  line  join- 
ing the  intersections  of  pairs  of 
opposite  sides,  viz.,  the  line 
XZ.  Similarly,  with  L  and  N 
as  centers,  we  can  show  that  the 
tangents  at  L  and  N  also  inter- 
sect on  XZ. 


and  K  and  M  the  points  of  con- 
tact on  k  and  m  respectively 
(Fig.  48).  The  envelope  is 
then  generated  by  the  two  pro- 
jectively  related  point-rows 
AKDF  and  BFCM  (Art. 
45).  When  we  project  these 
point-rows  from  B  and  A, 
respectively,  we  obtain  the 
two  sheaves  B(AKDF)  and 
A(BFCM)  which  are  in  per- 
spective position,  since  they 
have  the  line  AB  as  a  self- 
corresponding  ray.  The  line 
KM  is  the  axis  of  perspec- 
tivity. Hence,  the  line  joining 
the  points  of  contact  on  the 
opposite  sides  k,  m  passes 
through  the  intersection  of  the 
lines  joining  opposite  vertices, 
viz.,  the  point  Y.  Similarly, 
with  I  and  n  as  point-rows,  we 
can  show  that  the  line  joining 
the  points  of  contact  on  I  and  n 
also  passes  through  Y. 


61.  Generation  of  Particular  Conies  and  Envelopes. — The  conic 
generated  by  two  projectively  related  sheaves  of  rays  will  be  an 
ellipse,  a  parabola,  or  a  hyperbola  according  as  the  sheaves  have 
no  pair,  one  pair,  or  two  pairs  of  corresponding  parallel  rays.  If 
one  of  the  sheaves  is  superposed  upon  the  other  in  such  a  way 
that  the  direction  of  its  rays  is  unchanged,  then  these  sheaves  will 
have,  in  the  first  case,  no  self-corresponding  rays;  in  the  second 
case,  one;  and  in  the  third  case,  two  such  rays.  If  the  sheaves 
are  oppositely  projective  (Art.  40),  they  will  always  generate  a 
hyperbola. 

The  envelope  generated  by  two  projectively  related  point-rows 
will  be  the  system  of  tangents  to  a  parabola  if  the  ideal  points  on 


§63]  PASCAL  AND  BRIANCHON  THEOREM  79 

the  point-rows  correspond  to  each  other,  for  then  the  ideal  line  is 
one  of  the  tangents  to  the  curve.  If  the  two  point-rows  are 
placed  in  perspective  position,  by  superposing  any  two  actual 
homologous  points  (Fig.  49),  they  will  be  sections  of  a  sheaf  of 
parallel  rays.  Corresponding  segments  along  the  two  point- 
rows  are  thus  seen  to  be  proportional  to  each  other.  Hence, 
any  two  tangents  to  a  parabola  are  cut  proportionally  by  the 
remaining  tangents.  For  this  reason,  protectively  related  point- 
rows  whose  ideal  points  correspond  to  each  other  are  called 
similarly  projective. 


FIG.  49. 


62.  Cones  and  Sheaves  of  Planes  of  Second  Class. — Since  the 
tangents  to  a  curve  of  second  order  form  a  sheaf  of  rays  of  second 
class,  it  follows  by  projection  that  the  tangent  planes  to  a  cone 
form  a  sheaf  of  planes  of  second  class. 

Corresponding  to  the  theorems  of  Pascal  and  Brianchon,  there 
are  correlative  theorems  for  the  cone  and  the  sheaf  of  planes  of 
second  class  which  are  at  once  derived  by  projecting  the  figures  for 
the  Pascal  and  Brianchon  theorems  from  a  point  outside  the  plane 
in  which  they  lie. 

63.  Cylinders. — A  cone  whose  vertex  is  an  ideal  point  is  a 
cylinder.     Two  sheaves  of  planes  which  are  projectively  related, 
but  not  in  perspective  position,  and  whose  axes  are  parallel  gen- 
erate a  cylinder. 

A  cylinder  is  elliptic,  parabolic,  or  hyperbolic  according  as  a 
plane,  not  passing  through  its  ideal  vertex,  cuts  it  in  an  ellipse,  a 
parabola,  or  an  hyperbola. 


80  PROJECTIVE  GEOMETRY  [§63 

Exercises 

1.  If  a  point-row  u  and  a  sheaf  of  rays  S  are  coplanar  and  pro- 
jectively  related,  and  through  each  point  of  u  is  drawn  a  straight  line 
parallel  to  the  corresponding  ray  of  S;  show  that  these  lines  will 
either  intersect  in  one  point  or  will  envelope  a  parabola. 

2.  Construct  a  hyperbola  of  which  there  are  given  the  asymptotes 
and  one  point  or  one  tangent. 

3.  Construct  a  parabola  of  which  there  are  given  four  tangents, 
or  three  tangents  and  the  point  of  contact  on  one  of  them,  or  two 
tangents  and  their  points  of  contact. 

4.  Given  a  point-row  u  and  a  sheaf  of  rays  S  which  are  coplanar 
and  projectively  related;  show  that  straight  lines  drawn  from  the 
points  of  u  perpendicular  to  the  corresponding  rays  of  S  will  either 
envelope  a  parabola  or  pass  through  one  point. 

6.  If  the  vertices  of  a  triangle  move  upon  three  fixed  lines  of  the 
plane  in  such  a  manner  that  two  sides  of  the  triangle  do  not  alter  their 
directions;  show  that  the  third  side  will  either  envelope  a  parabola  or 
move  parallel  to  itself. 

6.  If  a  triangle  is  inscribed  in  a  conic,  show  by  the  Pascal  theorem 
that  the  tangents  at  the  vertices  meet  the  opposite  sides  in  three 
collinear  points  (triangle  theorem). 

7.  If  a  triangle  is  circumscribed  about  sL  conic,  show  that  the  lines 
joining  each  vertex  to  the  point  of  contact  on  the  opposite  side  are 
concurrent  (dual  of  the  triangle  theorem). 

8.  Prove  the  pentagon  theorem  without  making  use  of  the  Pascal 
theorem. 

9.  If  a  hexagon  whose  vertices  are  not  coplanar  nor  its  three  diag- 
onals concurrent  is  projected  from  any  point  on  a  line  which  meets  all 
three  of  the  diagonals,  show  that  the  lines  projecting  the  vertices  are 
rays  of  a  cone  of  second  order. 

NOTE. — The  Pascal  theorem  was  discovered  by  Pascal  when  he  was 
only  16  years  of  age.  It  was  first  published  in  a  little  work  entitled 
Essai  pour  les  coniques  in  which  it  was  called  the  theorem  of  the  mystic 
hexagram  (1640).  Pascal  proved  his  theorem  first  for  the  circle  and 
then,  by  means  of  projection,  extended  it  to  any  conic. 

Brianchon  discovered  the  dual  theorem  in  1806  and  published  it  in 
the  Journal  de  I'Ecole  poly  technique.  This  was  before  the  principle 
of  duality  had  been  enunciated  by  Poncelet  and  Gergonne. 


CHAPTER  VIII 

POLES  AND  POLAR  LINES  WITH  RESPECT  TO  A  CURVE 
OF  SECOND  ORDER 

64.  Poles  and  Polar  Lines. — The  theory  of  poles  and  polar  lines 
with  respect  to  a  fixed  conic  is  based  upon  the  quadrangle  theo- 
rem and  its  dual  (Art.  57).  The  following  dual  theorems  are 
fundamental. 


Theorems  X. — 

//  X  is  a  point  in  the  plane  of  a 
fixed  conic,  the  harmonic  con- 
jugate of  X,  with  respect  to  the 
curve  points  on  any  secant  drawn 
through  X,  lies  on  a  fixed  straight 
line  called  the  polar  line  of  X 
with  respect  to  the  conic. 


If  x  is  a  line  in  the  plane  of  a 
fixed  conic,  the  harmonic  con- 
jugate of  x,  with  respect  to  the 
tangents  to  the  conic  from  any 
point  of  x,  passes  through  a 
fixed  point  called  the  pole  of  x 
with  respect  to  the  conic. 


Thus,  in  Fig.  50,  LM  and  KN  are  any  two  secants  drawn  through 
X.  Pairs  of  opposite  sides  of  the  inscribed  quadrangle  KLMN 
meet  in  the  points  X,  Y,  Z.  The  harmonic  range  MVLX  is 
defined  by  the  quadrangle  ZK  YN,  and  the  harmonic  range  N  UKX, 
by  the  quadrangle  ZLYM.  Hence,  the  harmonic  conjugates  of  X 
with  respect  to  the  curve  points  on  the  secants  KN  and  LM  lie 
on  the  line  ZY.  We  now  have  to  show  that  this  line  is  fixed  in 
position  irrespective  of  the  secants  drawn  through  X.  To  do  this, 
we  recall  that  the  tangents  at  opposite  vertices  of  the  simple  quad- 
rangle KMNL  intersect  on  YZ;  that  is,  the  points  A  and  B  lie  on 
YZ  (Art.  57).  Hence,  if  we  fix  the  secant  LM  and  thus  fix  the 
points  A  and  V,  the  line  YZ  is  fixed  irrespective  of  the  position  of 
6  81 


82 


PROJECTIVE  GEOMETRY 


[§64 


the  secant  XKN.  The  harmonic  conjugate,  U,  must  then  lie  on 
the  fixed  line  YZ,  whatever  position  is  taken  by  the  secant  XKN. 
Or,  if  we  fix  the  secant  KN,  the  fixed  points  B  and  U  determine 


FIG.  50a. 

the  position  of  the  line  YZ;  and  V,  the  harmonic  conjugate  of  X 
with  respect  to  L  and  M,  lies  on  YZ  whatever  position  is  taken  by 

the  secant  XLM.  Therefore  the 
position  of  the  line  YZ  is  inde- 
pendent of  the  secants  drawn 
through  X. 

For  the  dual  theorem,  A  and 
B  are  any  two  points  on  x  from 
which  tangents  can  be  drawn  to 
the  conic.  Let  K,  L,  M,  N  be  the 
points  of  contact  of  these  tan- 
gents; then  (Art.  57)  the  lines 
KM  and  LN  intersect  on  the 
diagonal  x  of  the  simple  circum- 
scribed quadrangle  whose  sides 
are  the  tangents  at  K,  L,  M,  N. 

Also,  the  lines  KL  and  NM  intersect  on  x,  since  a;  is  a  diagonal  of 
the  quadrangle  formed  by  the  tangents  at  K,  N,  L,  M.  The 
lines  KN  and  LM  intersect  in  a  point  X  through  which  pass  the 


FIG.  506. 


§66]  POLES  AND  POLAR  LINES  83 

harmonic  conjugates  of  x  with  respect  to  the  tangents  from  A 
and  from  B,  since  the  pencils  B(NUKX)  and  A(MVLX)  are 
harmonic.  Hence,  if  B  is  fixed,  X  is  determined  by  the  fixed 
line  NK  and  the  harmonic  conjugate  of  x  with  respect  to  the 
tangents  from  B  and  if  A,  is  fixed,  X  is  determined  by  the  fixed 
line  ML  and  the  harmonic  conjugate  of  x  with  respect  to  the 
tangents  from  A.  Consequently,  the  position  of  X  is  independent 
of  the  positions  of  A  and  B  upon  x. 

65.  Special  Positions  of  Pole  and  Polar  Line. — Since  a  point  X 
is  separated  harmonically  from  points  of  its  polar  line  by  the  conic, 
it  follows  that,  if  X  is  inside  the  curve  (Fig.  50  (a)),  its  polar  line 
lies  wholly  outside  the  curve;  and  if  X  is  outside  the  conic  (Fig. 
50  (6)),  its  polar  line  cuts  the  curve. 

Again,  if  X  lies  on  the  conic,  all  the  harmonic  conjugates  of  X 
with  respect  to  curve  points  coincide  with  X  (cf.  Art.  37).  The 
polar  line  of  X  is  then  the  tangent  to  the  conic  at  X. 

Similarly,  if  a  line  x  does  not  meet  the  fixed  conic,  its  pole 
lies  inside  the  curve.  For  then  the  harmonic  conjugates  of 
x,  with  respect  to  tangents  drawn  from  points  of  x,  all  meet 
the  curve.  But  if  x  meets  the  conic,  none  of  these  harmonic 
conjugates  meets  the  curve,  and  the  pole  is  consequently  out- 
side the  conic. 

If  x  touches  the  conic,  the  harmonic  conjugate  of  x  with  respect 
to  the  tangents  from  any  point  of  x  coincides  with  x,  since  x  is  one 
of  the  two  tangents.  The  pole  is  then  the  point  of  contact  of  x 
with  the  conic. 

In  conclusion,  therefore,  we  can  say: 

With  respect  to  a  fixed  conic,  every  point  in  the  plane  has  a  definite 
polar  line;  and  every  line  in  the  plane  has  a  definite  pole. 

66.  Chords  of  Contact. — If  X  is  a  point  outside  a  conic,  the  line 
joining  the  points  of  contact  of  tangents  from  X  is  called  the 
chord  of  contact  of  these  tangents. 

If  in  Fig.  50  (6),  the  secant  KN  is  allowed  to  rotate  about  X 
until  the  curve  points  coincide,  the  secant  then  becomes  a  tangent 
to  the  conic  from  X.  But  the  curve  points  can  only  coincide  on 
the  polar  line  of  X,  since  XKUN  is  constantly  a  harmonic  range. 
Similarly,  the  curve  points  L  and  M  can  only  coincide  on  the  polar 
line.  Hence: 


84  PROJECTIVE  GEOMETRY  [§67 


//  X  is  a  point  outside  a  conic, 
its  polar  line  with  respect  to  the 
conic  is  the  chord  of  contact  of 
the  tangents  drawn  from  X. 


If  x  cuts  a  conic,  its  pole  with 
respect  to  the  conic  is  the  inter- 
section of  the  tangents  at  the 
points  in  which  x  meets  the 


come. 


67.  Construction  of  Poles  and  Polar  Lines. — If  a  conic  is  fully 
drawn,  the  fundamental  theorems  in  Art.  64  will  serve  to  determine 
the  polar  line  of  a  given  point  with  respect  to  the  conic,  or  the  pole 
of  a  given  line  with  respect  to  the  conic.  Thus  (Fig.  50),  with  X  as 
the  given  point,  the  pairs  of  points  U  and  V,  or  Y  and  Z,  or  A  and 
B  can  be  constructed.  Any  one  of  these  pairs  of  points  deter- 
mines the  polar  line  of  X. 

With  x  as  the  given  line,  the  pairs  of  lines  AX  and  BX,  or  a  pair 
of  diagonals  of  a  circumscribed  quadrilateral,  one  of  whose  diago- 
nals is  x,  or  the  lines  LM  and  KN  can  be  constructed.  Any  one 
of  these  pairs  of  lines  determines  the  pole  of  x. 

If  the  given  point  is  outside  the  conic  (the  given  line  cuts 
the  conic),  Art.  66  may  be  used  to  construct  the  polar  line 
(the  pole). 

If  a  conic  is  not  fully  drawn,  but  is  given  by  any  one  of  the  sets 
of  conditions  in  Art.  46,  the  polar  line  of  a  given  point  with  respect 
to  the  conic  (or  the  pole  of  a  given  line)  can  be  constructed  with 
the  aid  of  a  ruler.  Thus,  if  only  five  points  of  a  conic  are  known, 
the  polar  line  of  any  given  point  can  be  constructed  as  follows: 
Join  the  given  point  to  any  two  of  the  known  points,  and,  with  the 
aid  of  the  Pascal  theorem,  construct  the  points  where  these  lines 
meet  the  conic  again  (Art.  53).  We  now  have  an  inscribed  quad- 
rangle two  of  whose  sides  pass  through  the  given  point  and  we  can 
complete  the  construction  by  Art.  64. 

If  only  five  tangents  to  a  conic  are  known,  we  can  construct 
the  points  of  contact  on  these  by  Art.  56  and  then  proceed  as 
before. 

If  only  five  tangents  to  a  conic  are  known,  we  can  construct  the 
pole  of  any  given  line  by  finding  the  points  of  intersection  of  this 
line  with  two  of  the  tangents  and  then  (Art.  53).  constructing  a 
complete  circumscribed  quadrilateral  one  of  whose  diagonals  is  the 
given  line.  The  other  two  diagonals  intersect  in  the  required  pole. 


POLES  AND  POLAR  LINES 


85 


If  the  conic  is  given  by  five  points,  we  can  construct  the  tangents 
at  these  (Art.  56)  and  proceed  as  before. 

If  any  particular  case,  the  general  procedure  outlined  above  is 
to  be  followed.  For  example,  suppose  it  is  required  to  construct 
the  polar  line  of  a  given  point  when  the  conic  is  given  by  three 
tangents  and  the  points  of  contact  on  two  of  them.  Let  a,  6,  and 


FIG.  51. 


c  be  the  three  tangents  and  A  and  B  the  points  of  contact  on  a  and 
6  respectively  (Fig.  51).  Construct  the  point  of  contact,  C,  on  c 
(Art.  63,  exercise  7).  Let  X  be  the  given  point  whose  polar  line  is 
required.  Join  X  to  two  of  the  points  of  contact,  say  A  and  C,  and 
determine  the  points  where  the  lines  XA  and  XC  meet  the  conic 


86  PROJECTIVE  GEOMETRY  [§68 

again;  that  is,  construct  the  inscribed  quadrangles  ABCD  and 
AECB.  In  the  first  quadrangle,  the  Pascal  line  is  determined  by 
the  intersection  of  the  lines  AB,  XC  and  the  intersection  of  the 
tangents  at  A  and  C.  The  sides  EC  and  AD  meet  on  this  line. 
Hence,  D  is  determined.  Similarly,  E  is  determined.  We  now 
have  an  inscribed  quadrangle  two  of  whose  sides  meet  in  the  given 
point  X,  viz.,  the  quadrangle  AEDC.  The  sides  AD,  CE  and  the 
sides  AC,  DE  intersect  on  the  required  polar  line. 


Exercises 

1.  Construct  the  polar  line  of  a  given  point  with  respect  to  a  conic 
which  is  given  by  five  tangents. 

2.  A  conic  is  fully  drawn,  construct  the  tangents  to  it  from  an  ex- 
terior point. 

3.  A  conic  touches  four  given  lines  one  of  which  is  the  ideal  line. 
The  point  of  contact  on  the  ideal  line  being  given,  construct  the 
polar  line  of  a  given  point;  the  pole  of  a  given  line. 

4.  A  conic  is  given  by  five  points  two  of  which  are  ideal,  construct 
the  pole  of  a  given  line;  the  polar  line  of  a  given  point. 

6.  A  conic  is  given  by  three  points  and  the  tangents  at  two  of  them, 
construct  the  pole  of  a  given  line. 

68.  Conjugate  Points  and  Conjugate  Lines  with  Respect  to  a 
Conic. — If,  in  the  plane  of  a  fixed  conic,  two  points  are  so  situated 
that  each  lies  on  the  polar  line  of  the  other,  the  points  are  said  to  be 
conjugate  points  with  respect  to  the  conic;  and  if  two  lines  are  so 
situated  that  each  passes  through  the  pole  of  the  other,  the  lines 
are  called  conjugate  lines  with  respect  to  the  conic. 

The  following  dual  theorems  are  fundamental  for  conjugate 
points  and  conjugate  lines. 

Theorems  XI. — 


//  one  point  lies  on  the  polar 
line  of  another,  the  two  points 


If  one  line  passes  through  the 
pole  of  another,  the  two  lines  are 


are  conjugate  with  respect  to  the      conjugate    with    respect    to    the 


come. 


conic. 


For  the  theorem  on  the  left,  if  Y  is  any  point  of  the  polar  line  of 


POLES  AND  POLAR  LINES  87 

A',  we  are  to  prove  that  the  polar  line  of  F  passes  through  X. 
There  are  three  cases  to  consider. 

1.  If  X  is  inside  the  conic  (Fig.  50),  it  is  separated  harmonically 
from  every  point  of  its  polar  line  (and  therefore  from  F)  by  the 
curve.     But  F  is  separated  harmonically  from  all  points  of  its 
polar  line  that  are  inside  the  conic  (and  from  no  other  points  inside 
the  conic)  by  the  curve.     Since  F  is  separated  harmonically  from 
X  by  the  curve,  and  X  is  inside  the  conic,  X  is  on  the  polar  line 
of  F.  . 

2.  If  A"  is  on  the  conic,  F  is  on  the  tangent  to  the  conic  at  X 
(Art.  65).     The  polar  line  of  F  is  the  chord  of  contact  of  tangents 
to  the  conic  from  F  and  consequently  passes  through  X. 

3.  If  X  is  outside  the  conic,  F  is  on  the  chord  of  contact  of  tan- 
gents from  X  (Art.  66).     But  the  tangents  at  the  curve  points  on 
any  secant  through  F  intersect  on  the  polar  line  of  F  (Art.  67). 
Therefore,  X  is  on  the  polar  line  of  F. 

Hence,  in  every  case,  the  polar  line  of  F  passes  through  X;  and 
X  and  F  are,  therefore,  by  definition,  conjugate  points  with  respect 
to  the  conic. 

For  the  theorem  on  the  right,  if  y  passes  through  the  pole  of  x, 
the  pole  of  y  lies  on  x  by  what  has  just  been  proved.  Therefore, 
x  and  y  are  conjugate  lines  with  respect  to  the  conic. 

69.  Consequences  of  Theorems  XI. — A  given  point  is  conjugate 
to  all  the  points  on  its  polar  line  and  a  given  line  is  conjugate  to  all 
the  lines  through  its  pole.  Hence,  if  a  point  describes  a  given 
line,  its  polar  line  will  rotate  about  the  pole  of  the  given  line. 
Likewise,  if  a  line  rotates  about  a  given  point,  its  pole  will  describe 
the  polar  line  of  the  given  point. 

Thus,  a  given  point-row  has  a  sheaf  of  polar  lines;  and  a  given 
sheaf  of  rays  has  a  point-row  of  poles. 

1.  A  point-row  x  is  projectively  related  to  its  sheaf  of  polar  lines  X. 

Thus,  in  Fig.  50,  let  X,  L,  and  M  be  fixed  while  N  describes  the 
curve.  The  points  Z  and  F  will  then  describe  projectively  related 
point-rows  along  the  fixed  line  x,  since  the  rays  MN  and  LN 
describe  projectively  related  sheaves  of  rays  about  M  and  L 
respectively  (Art.  45,  3).  But  the  point-row  described  by  F  is 
projected  from  X  by  the  sheaf  of  rays  described  by  AT  which 
is  the  polar  line  of  Z.  Hence,  the  point-row  described  by  Z  is 


88 


PROJECTIVE  GEOMETRY 


[§69 


projectively  related  to  the  sheaf  of  rays  described  by  the  polar 
line  of  Z. 


2.  If  U  and  V  are  two  non- 
conjugate  points  in  the  plane,  for 
every  ray  p  of  U  there  is  a  ray  p\ 
of  V  conjugate  to  it.  The  two 
sheaves  of  rays  thus  constructed 
are  projectively  related  and  gen- 
erate a  curve  of  second  order,  or 
a  point-row  of  first  order,  accord- 
ing as  the  line  UV  does  not,  or 
does,  touch  the  fixed  conic. 


If  u  and  v  are  two  non-conju- 
gate lines  in  the  plane,  for  every 
point  P  of  u  there  is  a  point  PI 
of  v  conjugate  to  it.  The  two 
point-rows  thus  constructed  are 
projectively  related  and  generate 
an  envelope  of  second  class,  or  a 
sheaf  of  rays  of  first  class,  accord- 
ing as  the  point  (uv)  does  not,  or 
does,  lie  on  the  fixed  conic. 


For,  on  the  left,  each  sheaf  is  the  projector  of  the  point-row  of 
poles  belonging  to  the  other;  and  on  the  right,  each  point-row  is 
a  section  of  the  sheaf  of  polar  lines  belonging  to  the  other. 


FIG.  52. 

3.  If  a  triangle  is  inscribed  in 
a  conic,  any  line  conjugate  to  one 
side  with  respect  to  the  conic  cuts 
the  other  two  sides  in  conjugate 
points;  and  conversely. 

For,  if  ABC  is  any  triangle 
inscribed  in  a  conic  and  S  is  the 
pole  of  AB  (Fig.  52),  then  the 


FIG.  53. 

//  a  triangle  is  circumscribed 
about  a  conic,  any  point  conju- 
gate to  one  vertex  with  respect  to 
the  conic  is  projected  from  the 
other  two  vertices  by  conjugate 
lines;  and  conversely. 

For,  if  abc  is  any  triangle  cir- 
cumscribed about  a  conic  and  s 
is  the  polar  line  of  (a&)  (Fig.  53), 


§70] 


POLES  AND  POLAR  LINES 


89 


sheaves  of  rays  A  (CBS — )  and 
B(CSA— )  are  project! vely  re- 
lated and  cut  the  sides  BC  and 
AC,  respectively,  in  point-rows 
which  are  in  perspective  posi- 
tion, S  being  the  center  of  per- 
spectivity.  Any  line  through 
S,  therefore,  cuts  these  point- 
rows  in  a  pair  of  corresponding 
points,  as  P  and  PI.  The  lines 
PB  and  P\A  intersect  on  the 
conic  and  form  sides  of  an  in- 
scribed quadrangle  ABCD. 
Hence,  the  polar  line  of  P  passes 
through  PI. 


then  the  point-rows  ABC —  and 
AiBiCi —  are  protectively  re- 
lated and  are  projected  from  the 
vertices  C\  and  C,  respectively, 
in  sheaves  of  rays  which  are  in 
perspective  position,  s  being  the 
axis  of  perspectivity.  Any 
point  on  s,  therefore,  is  pro- 
jected from  C  and  Ci  by  a  pair 
of  corresponding  rays,  as  p  and 
Pi.  The  points  D  and  DI  are 
joined  by  a  tangent  to  the  conic 
and  form  vertices  of  a  circum- 
scribed quadrilateral  DCCiDi. 
Hence,  the  pole  of  p  lies  on  p\. 


70.  Polar  Figures  with  Respect  to  a  Fixed  Conic. — By  virtue  of 
Art.  65  and  with  the  aid  of  theorems  XI,  we  can  say  that,  with 
respect  to  a  fixed  conic,  any  plane  figure  consisting  of  points  and 
lines  has  a  polar  figure  consisting  of  lines  and  points.  Thus,  a 
triangle  ABC  has  a  polar  triangle  whose  sides  are  the  polar  lines 
of  A,  B,  and  C  and  whose  vertices  are  the  poles  of  the  sides  of  ABC. 

Again  (Art.  69,  1),  we  see  that  the  polar  of  a  harmonic  range  of 
points  is  a  harmonic  sheaf  of  rays. 

If  we  regard  a  plane  curve  as  described  by  a  moving  point,  the 
polar  line  of  the  point  will  envelope  the  polar  curve.  Thus,  if  a 
point  describes  a  curve  of  second  order,  its  polar  line  will  envelope 
a  curve  of  second  class.  For  the  point  is  the  intersection  of  cor- 
responding rays  in  two  protectively  related  sheaves  of  rays  and 
hence  its  polar  line  joins  corresponding  points  in  two  projectively 
related  point-rows. 

A  fixed  conic  may  thus  be  thought  of  as  polarizing  the  plane  in 
which  it  lies  in  such  a  way  that  to  every  line  corresponds  a  definite 
point  and  to  every  point  corresponds  a  definite  line;  and,  further, 
if  a  point  lies  on  a  line,  the  corresponding  line  passes  through  the 
corresponding  point. 

As  another  example,  consider  a  simple  hexagon  inscribed  in  a 
conic  (Fig.  54).  The  polar  figure  with  respect  to  the  conic  is  a 
simple  circumscribed  hexagon,  since  the  polar  line  of  a  point  on 
the  conic  is  the  tangent  at  that  point;  and  the  pole  of  any  chord 
is  the  intersection  of  the  tangents  at  the  extremities  of  the  chord. 


90 


PROJECTIVE  GEOMETRY 


A  pair  of  opposite  sides  of  the  inscribed  hexagon  intersect  in  a 
point  whose  polar  line  joins  the  corresponding  opposite  vertices  of 
the  circumscribed  hexagon.  Since  pairs  of  opposite  sides  of  the 
inscribed  hexagon  intersect  in  collinear  points  (Pascal  theorem), 
it  follows  that  pairs  of  opposite  vertices  of  the  circumscribed  hexa- 
gon are  joined  by  concurrent  lines  (Brianchon  theorem).  Thus 
either  the  Pascal  theorem  or  the  Brianchon  theorem  can  be  derived 


FIG.  54. 


one  from  the  other,  by  means  of  polar  figures  with  respect  to  a 
fixed  conic.  The  Pascal  line  thus  appears  as  the  polar  line  of  the 
Brianchon  point. 

The  concept  of  polar  figures  with  respect  to  a  fixed  conic  is  but 
a  special  case  of  the  concept  of  dual,  or  reciprocal,  figures  in  the 
plane.  Polar  figures  may  thus  be  thought  of  as  dual,  or  reciprocal, 
figures  with  respect  to  a  fixed  conic.  With  respect  to  a  fixed 
conic,  the  dual,  or  reciprocal,  of  a  given  figure  is  perfectly  definite, 
and  can  be  constructed,  while  the  general  principle  of  duality,  as 


§72]  POLES  AND  POLAR  LINES  91 

we  know,  merely  asserts  the  existence  of  a  dual  figure  without 
stating  laws  for  its  construction. 

71.  Self-polar  Figures. — If  a  plane  figure  coincides  with  its 
polar  figure,  it  is  called  self -polar.  Thus  two  polar  triangles,  con- 
sidered as  a  single  figure,  form  a  self-polar  figure. 

Again  (Fig.  55),  the  triangle  XYZ  is  self -polar,  since  the  polar 
line  of  any  vertex  is  the  side  opposite  that  vertex.  The  tangents 


M 


FIG.  55. 

at  K,  L,  M,  N  intersect  in  pairs  on  the  sides  of  the  triangle  XYZ; 
and  hence  the  sides  of  XYZ  are  the  diagonals  of  the  complete 
circumscribed  quadrilateral  formed  by  these  tangents.  For  this 
reason,  the  triangle  XYZ  is  called  a  diagonal  triangle. 

Opposite  sides  of  the  complete  inscribed  quadrangle  KLMN 
intersect  in  the  vertices  of  the  diagonal  triangle;  and  hence  the 
diagonal  triangle  is  completely  determined  by  the  vertices  of  the 
quadrangle.  The  diagonal  triangle  is  thus  self-polar  with  respect 
to  any  conic  passing  through  K,  L,  M,  and  N. 

Any  two  points  X  and  Z,  conjugate  with  respect  to  a  conic, 
are  vertices  of  a  diagonal  triangle;  the  third  vertex  is  the  intersec- 
tion of  the  polar  lines  of  X  and  Z. 

72.  Pole-rays  and  Polar  Planes  with  Respect  to  a  Cone. — If 
we  project  a  conic  from  a  point  not  lying  in  its  plane  we  obtain 
a  cone;  and  the  properties  we  have  been  developing  with  refer- 
ence to  the  conic  go  over  into  properties  connected  with  the  cone. 
Thus,  by  this  projection,  pole  and  polar  line  with  respect  to  the 


92  PROJECTIVE  GEOMETRY  [§72 

conic  become  pole-ray  and  polar  plane  with  respect  to  the  cone. 
If  a  is  any  line  through  the  vertex  of  the  cone,  and  we  draw  any 
plane  through  a  cutting  the  cone  in  the  rays  b  and  d  and  the  polar 
plane  of  a  in  c,  then  abed  is  a  harmonic  pencil  of  rays. 

The  projector  of  a  pair  of  polar  figures  with  respect  to  the 
conic  is  a  pair  of  polar  figures  with  respect  to  the  cone. 

The  projector  of  a  diagonal  triangle  is  a  diagonal  pyramid. 
The  faces  of  a  diagonal  pyramid  are  the  diagonal  planes  of  a 
complete  4-face  circumscribed  about  the  cone;  and  the  edges 
of  a  diagonal  pyramid  are  the  intersections  of  pairs  of  opposite 
faces  of  a  complete  inscribed  4-edge. 

If  a  plane  touches  a  cone,  its  pole-ray  is  the  line  of  contact  of 
the  plane;  and  the  polar  plane  of  a  ray  of  the  cone  is  the  tangent 
plane  along  that  ray. 

A  cone  may  be  thought  of  as  polarizing  the  bundle  of  rays  whose 
center  is  its  vertex  in  such  a  way  that  to  every  ray  of  the  bundle 
corresponds  a  definite  plane  (polar  plane)  and  to  every  plane  of  the 
bundle  corresponds  a  definite  ray  (pole-ray). 


Exercises 

1.  Construct  the  polar  figure  of  a  given  triangle  one  of  whose  sides 
is  the  ideal  line. 

2.  If  two  tangents  to  a  conic  vary  so  that  their  chord  of  contact 
envelopes  a  second  conic,  show  that  their  intersection  will  trace  a 
third  conic;  and  conversely. 

3.  Two   conies  intersect  in  four  points,   construct   the   diagonal 
triangle  common  to  both. 

4.  Two  conies  intersect  in  two  points,  construct  one  vertex  and  the 
side  opposite  of  the  diagonal  triangle  common  to  both. 

5.  If  two  conies  do  not  intersect  each  other,  construct  the  diagonal 
triangle  common  to  both. 

6.  Given  two  conies  in  the  plane.     Any  point  in  the  plane,  as  A, 
has  a  polar  line  with  respect  to  each  of  them,  and  if  these  polar  lines 
intersect  in  Ai,  A  and  A\  are  conjugate  with  respect  to  both  curves. 
Show  that  if  A  describes  a  straight  line,  Ai  will,  in  general,  describe  a 
third  conic  which  passes  through  the  vertices  of  the  diagonal  triangle 
common  to  the  two  given  conies. 

NOTE. — The  theory  of  poles  and  polar  lines  is  due  to  Desargues, 


§72]  POLES  AND  POLAR  LINES  93 

whose  development  of  the  theory  is  contained  in  his  Brouillon  projet 
d'une  atteint — (1639).  Earlier  writers  had  discovered  certain  theo- 
rems and  properties.  Apollonius,  for  example,  knew  that  the  inter- 
section of  two  tangents  to  a  conic  is  harmonically  separated  from  the 
chord  of  contact  by  the  curve  points  on  any  secant  through  the 
intersection. 

The  correlation  in  exercise  6  is  known  as  a  Steiner  Correlation.     It 
belongs  to  the  theory  of  quadric  transformations. 


CHAPTER  IX 

DIAMETERS,    AXES,   AND    ALGEBRAIC    EQUATIONS    OF 
CURVES  OF  SECOND  ORDER 

73.  Diameters  and  Centers  of  Conies. — The  theory  of  poles 
and  polar  lines  with  respect  to  a  conic  leads  to  important  metric 
properties  of  the  conic.  Thus,  the  polar  line  of  an  ideal  point 
is  a  diameter  of  the  conic;  and  the  pole  of  the  ideal  line  is  the 
center  of  the  conic. 

Every  diameter  of  a  conic  passes  through  the  center  (Art.  69); 
and  the  segment  between  the  curve  points  on  each  diameter  is 
bisected  by  the  center  (Art.  64).  Hence  (Art.  65),  the  center 
of  an  ellipse  is  inside  the  curve;  the  center  of  a  hyperbola  is 
outside  the  curve;  and  the  center  of  a  parabola  is  the  point  of 
contact  of  the  curve  with  the  ideal  line.  The  diameters  of  a 
parabola,  therefore,  form  a  system  of  parallel  lines. 

Again  (Art.  64):  Any  diameter  of  a  conic  bisects  a  system  of 
parallel  chords;  namely,  all  the  chords  which,  when  extended,  pass 
through  the  infinitely  distant  pole  of  the  diameter. 


FIG.  56.  FIG.  57. 

74.  Conjugate  Diameters.^Two  diameters  of  a  conic,  each 
of  which  passes  through  the  pole  of  the  other,  are  called  conjugate 
diameters  with  respect  to  the  conic.  Hence,  by  the  preceding 
article1.  Each  of  two  conjugate  diameters  bisects  all  the  chords  of 
the  conic  drawn  parallel  to  the  other. 

94 


§75]  EQUATIONS  OF  CURVES  95 

A  pair  of  conjugate  diameters  together  with  the  ideal  line  form 
the  sides  of  a  diagonal  triangle  (Art.  71).  It  follows  from  the 
definition  of  a  diagonal  triangle  that :  The  diagonals  of  a  parallelo- 
gram circumscribed  about  a  conic  are  conjugate  diameters  with 
respect  to  the  conic  (Fig.  56). 

Again:  The  sides  of  a  parallelogram  inscribed  in  a  conic  are 
parallel  to  a  pair  of  conjugate  diameters  (Fig.  57). 

It  follows  at  once  that,  if  any  point  on  a  conic  is  joined  to  the 
extremities  of  a  diameter,  the  two  chords  so  formed  are  parallel 
to  a  pair  of  conjugate  diameters. 

Exercises 

1.  Suppose  a  conic  is  given  by  four  points  and  the  tangent  at  one 
of  them,  construct  a  diameter  and  determine  the  center. 

2.  Draw  the  chord  of  a  given  conic  which  is  bisected  at  a  given  point. 

3.  Prove  that  the  chords  of  a  given  conic  which  are  bisected  by  any 
given  chord  envelope  a  parabola. 

Suggestion.— Use  1,  Art.  69,  to  show  that  the  given  chord  is  pro- 
jectively  related  to  the  ideal  line  in  such  a  way  that  any  bisected  chord 
joins  corresponding  points. 

4.  Construct  a  conic  having  given: 

(a)  Two  points  and  one  pair  of  conjugate  diameters; 
(6)   Two  tangents  and  one  pair  of  conjugate  diameters; 

(c)  Three  points  and  the  center; 

(d)  Three  tangents  and  the  center; 

(e)  One  point  and  two  pairs  of  conjugate  diameters; 
(/)    One  tangent  and  two  pairs  of  conjugate  diameters. 

6.  Construct  a  parabola  having  given  either  three  points  or  three 
tangents  and  the  direction  of  its  diameters. 

75.  Application  of  the  Harmonic  Properties  of  Poles  and  Polar 
Lines. — As  special  cases  of  the  general  definitions  and  theorems 
in  Art.  64,  we  have  the  following  propositions: 

1.  A  pair  of  conjugate  diameters  of  a  hyperbola  is  harmonically 
separated  by  the  asymptotes. 

For  the  pencil  in  question  consists  of  two  tangents  and  a 
pair  of  conjugate  lines  drawn  from  the  intersection  of  the  tangents. 

2.  The   chord  determined    upon    any  secant  of  a  hyperbola  is 
bisected  by  the  diameter  conjugate  to  the  secant  (Art.  73). 


96 


PROJECTIVE  GEOMETRY 


[§75 


By  1,  the  segment  of  the  secant  contained  between  the  asymp- 
totes is  also  bisected  by  the  conjugate  diameter.  Hence: 

3.  The  two  segments  of  any  secant  which  lie  between  a  hyperbola 
and  its  asymptotes  are  equal  in  length. 

This  proposition  furnishes  a  neat  construction  of  a  hyperbola 
when  the  asymptotes  and  one  point  of  the  curve  are  given. 
Thus  (Fig.  58),  let  OA  and  OAi  be  the  given  asymptotes,  and  P 


F>> 


FIG.  58. 

be  any  point  on  the  curve.  Draw  a  series  of  rays  through  P 

meeting  OA  in  B,  C,  D, ,  and  OAi  in  B\,  C\,  DI, .  Lay 

off  segments  BiQ,  C\R,  DiS, equal  respectively  to  the  seg- 
ments BP,  CP,  DP, .  Then  the  points  Q,  R,  S, are 

points  on  the  hyperbola.  Since  0  is  the  center  of  the  curve,  the 

points  Pi,  Qi,  Ri,  Si, ,  symmetrical  to  P,  Q,  R,  S, with 

respect  to  0,  also  lie  on  the  curve. 

4.  The  segment  of  any  tangent  to  a  hyperbola  contained  between 
the  asymptotes  is  bisected  by  the  point  of  contact. 

For,  if  one  of  two  conjugate  diameters  is  parallel  to  a  tangent, 
the  other  passes  through  the  point  of  contact. 


§76] 


EQUATIONS  OF  CURVES 


97 


5.  The  segment  of  the  line  contained   between  the  pole  of  any 
chord  of  a  parabola  and  the  mid-point  of  the  chord  is  bisected  by 
the  curve. 

For  the  line  in  question  is  a  diameter  of  the  parabola,  and  the 
two  curve  points  on  it  harmonically  separate  the  pole  from  the 
mid-point  of  the  chord. 

Exercises 

1.  Given  five  points,  two  of  which  are  ideal;  construct  the  asymp- 
totes of  the  hyperbola  determined  by  the  given  points.     Construct  a 
series  of  points  of  the  curve. 

2.  Given  the  asymptotes  of  a  hyperbola  and  one  other  tangent; 
construct  a  series  of  points  of  the  curve. 

3.  Given  five  points  in  a  plane;  construct  a  pair  of  conjugate 
diameters  of  the  curve  determined  by  the  given  points. 

4.  Given  five  lines  in  a  plane;  construct  the  center  of  the  curve  to 
which  the  five  lines  are  tangent. 

6.  Show  that  the  perpendiculars  dropped  from  any  point  S  upon 
the  diameters  of  a  given  conic  meet  the  conjugate  diameters  in  points 
of  a  hyperbola  which  passes  through  S  and  through  the  center  of  the 
given  conic. 

76.  The  Axes  of  a  Conic. — We  have  seen  (Art.  73)  that  a  diame- 
ter of  a  conic  bisects  a  system  of  parallel  chords.  Each  of  these 


FIG.  59. 

chords  is  conjugate  to  the  diameter,  since  each  passes  through  the 
pole  of  the  diameter.  Any  diameter,  therefore,  has  a  system  of 
parallel  conjugate  chords. 

If  a  diameter  is  perpendicular  to  its  system  of  parallel  chords, 
the  diameter  is  called  an  axis  of  the  conic. 

The  following  considerations  establish  the  existence  of  at  least 

7 


98 


PROJECTIVE  GEOMETRY 


[§76 


one  axis  for  each  conic  and  exhibit  the  method  for  constructing 
axes  when  the  curve  is  fully  drawn. 

First,  suppose  the  conic  is  an  ellipse  or  a  hyperbola  (Fig.  59). 
Construct  the  center  0,  the  pole  of  the  ideal  line.  The  diameters 
of  the  given  conic  are  evidently  not  all  of  the  same  length,  for 
then  the  conic  would  be  a  circle.  Let  AB  be  a  diameter  which 
is  neither  the  shortest  nor  the  longest  of  the  diameters.  The  circle 
on  AB  as  diameter  must  then  meet  the  given  conic  in  four  points 


FIG.  61. 


of  which  A  and  B  are  two.  We  can  now  construct  a  rectangle 
inscribed  in  the  given  conic.  The  diameters  of  the  conic  parallel 
to  the  sides  of  this  rectangle  are  axes. 

Second,  suppose  the  given  conic  is  a  parabola  (Fig.  60).  Con- 
struct any  diameter,  as  PQ,  the  polar  line  of  an  ideal  point.  Draw 
the  chord  DE  perpendicular  to  PQ  and  let  F  be  its  mid-point. 
The  line  VF,  parallel  to  PQ,  is  an  axis. 

It  thus  appears  that  an  ellipse,  or  a  hyperbola,  has  at  least  two 
axes  perpendicular  to  each  other;  and  a  parabola  has  at  least  one 
axis. 

We  can  now  show  that  an  ellipse  or  a  hyperbola  cannot  have 
more  than  two  axes;  and  a  parabola  cannot  have  more  than  one 
axis.  For,  if  a  and  b  are  two  axes  of  an  ellipse  or  of  a  hyperbola 
(Fig.  61),  and  if  P  is  any  point  of  the  curve,  we  can  construct  the 
rectangle  PQRS  inscribed  in  the  curve.  If  c  is  a  third  axis,  we  can 
also  construct  the  rectangle  PVRU  inscribed  in  the  curve.  But 


§77]  EQUATIONS  OF  CURVES  99 

the  circle  whose  center  is  0  and  whose  radius  is  OP  passes  through 
the  six  points  P,  V,  Q,  R,  U,  S  and  consequently  has  six  points  in 
common  with  the  given  curve.  The  circle,  therefore,  coincides 
with  the  given  conic  (Art.  46).  In  other  words,  if  a  conic  has  more 
than  two  axes,  it  is  necessarily  a  circle,  and  then  every  diameter  is 
an  axis. 

In  Fig.  62,  let  a  be  an  axis  of  a  parabola  and  P  be  any  point  of 
the  curve.     Then  a  must  bisect  the  perpen- 
dicular chord  PR.     If  c  is  also  an  axis,  c  is 

parallel  to  a  and  must  bisect  the  perpendicular     

chord  PS.  Thus  the  straight  line  PR  meets 
the  parabola  in  the  three  points  P,  S,  R. 
But  this  is  impossible.  Hence,  a  parabola 
can  have  but  one  axis.  FIG.  62. 

In  conclusion,  we  have  shown  that: 

An  ellipse,  or  a  hyperbola,  has  exactly  two  mutually  perpendicular 
axes;  and  a  parabola  has  but  one  axis. 

It  follows,  from  the  construction,  that  the  axes  of  an  ellipse,  or 
of  a  hyperbola,  are  conjugate  diameters  of  the  conic. 

77.  The  Vertices  of  a  Conic. — The  points  in  which  an  axis  meets 
a  conic  are  called  vertices  of  the  conic. 

An  ellipse  has  four  vertices  since  every  diameter,  and  conse- 
quently each  axis,  meets  the  curve  in  two  points. 

A  hyperbola  has  but  two  vertices  since  its  center  lies  outside 
the  curve  and  any  two  conjugate  diameters,  and  consequently  the 
axes,  are  separated  harmonically  by  the  asymptotes  (Art.  75,  1). 
Thus,  but  one  of  the  axes  meets  the  curve. 

A  parabola  has  but  one  actual  vertex  since  every  diameter,  and 
therefore  the  axis,  meets  the  curve  in  its  ideal  center. 

Exercises 

1.  Construct  a  series  of  points  on  a  parabola,  having  given  two 
points  or  two  tangents  and  the  axis. 

2.  Given  four  tangents  to  a  parabola;  construct  its  axis  and  vertex. 

3.  Construct  a  series  of  points  on  a  conic,  having  given  three  points 
or  three  tangents  and  one  axis. 

4.  Through  a  point  S  in  the  plane  of  a  given  conic  lines  are  drawn 
parallel  to  the  diameters  of  the  conic.     Show  that  the  sheaf  S  is  pro- 
jectirely  related  to  the  sheaf  of  conjugate  diameters,  the  two  sheaves 


100 


PROJECTIVE  GEOMETRY 


[§78 


generating  a  second  conic.  How  are  the  axes  of  the  second  conic 
related  to  the  axes  of  the  given  conic? 

6.  Show  that  the  axes  of  an  hyperbola  bisect  the  angles  formed  by 
the  asymptotes. 

78.  Algebraic  Equations  of  the  Conies. — Algebraic  equations 
for  the  several  conies  are  derived  from  special  cases  of  the  quad- 
rangle theorem  or  its  dual  (Art.  57). 

1.  The  Hyperbola. — In  Fig.  63,  let  the  asymptotes  be  the  coor- 
dinate axes,  and  let  P  be  any  point  on  the  curve  whose  coordinates 
are  x  =  OQ  and  y  =  QP.  The  tangent  at  P  meets  the  asymptotes 
in  K  and  N,  and  the  segment  KN  is  bisected  at  P  (Art.  75,  4). 
Hence  ON  =  2x  and  OK  =  2y. 


FIG.  63. 

If  LM  is  a  second  tangent  meeting  the  asymptotes  in  L  and  M, 
the  quadrangle  KLMN  circumscribes  the  curve,  and  its  diagonals, 
KM  and  LN,  are  parallel  since  they  must  meet  on  the  ideal  line 
(Art.  57).  The  two  triangles  LKN  and  NLM  are  therefore  of 
equal  area.  If  from  each  of  these  triangles  we  subtract  their 
common  part,  namely,  the  triangle  LON,  we  obtain  the  triangles 
KON  and  LOM  also  equal  in  area.  But  the  area  of  KON  is 

}i  ON  OK  sin  NOK  =  2xy  sin  NOK. 

For  all  positions  of  P  this  area  is  constantly  equal  to  the  area  of  the 
fixed  triangle  LOM.  Therefore : 

The  algebraic  equation  of  a  hyperbola,  referred  to  its  asymptotes  as 
coordinate  axes,  is  xy  =  k,  k  being  a  constant. 

2.  The  Ellipse. — Choose  for  coordinate  axes  a  pair  of  conjugate 
diameters,  as  .AJS'and  QR  (Fig.  64).  Let  P  be  any  point  on  the 


§78] 


EQUATIONS  OF  CURVES 


ellipse  whose  coordinates  are  x  =  OD  and  y  =  DP.  The  tangents 
at  the  opposite  vertices  of  the  inscribed  quadrangle  APBQ  must 
meet  on  the  line  ST  (Art.  57).  But  the  tangents  at  A  and  B  are 
parallel  to  QR.  Hence  ST  is  parallel  to  QR.  We  have,  then,  a 
series  of  similar  triangles  from  which  we  derive, 


AD      AH 
DP  ~  HS' 


and 


AH 
HT 


AO 
OQ 


DP      HT 

OB      HB 
OQ  ~  HS' 


(1) 
(2)" 


FIG.  64. 

If  the  lengths  of  the  semi-diameters  AO  and  OQ  are  represented 
by  a  and  b  respectively,  we  have  from  (1), 

a  +  x      AH     a-x      HB 
y      ~  HS'        y      ~  HT' 
and  from  (2), 

AH      HB      a 
HT  ~  HS  ~  b' 
Hence,  from  (3)  by  multiplication, 


(3) 


102 


PROJECTIVE  GEOMETRY 


[§78 


from  which  we  get, 


AH  HJB  _  a2 
HS'HT'b2' 


H- 


Therefore:  The  equation  of  an  ellipse,  referred  to  a  pair  of  con- 
jugate diameters  as  coordinate  axes,  is 


where  a  and  b  are  the  lengths  of  the  semi-diameters. 

In  this  connection,  it  is  worth  noting  that,  as  P  describes  the 
ellipse,  the  points  S  and  T  describe  point-rows  along  the  fixed  lines 


FIG.  65. 

BQ  and  QA  in  perspective  position,  since  the  line  ST  is  constantly 
parallel  to  QR.  The  sheaves  of  rays  described  by  AS  and  BT  are, 
therefore,  protectively  related  and  generate  the  ellipse.  The  chain 
of  perspectivity 

A  A  BQ  A  QA  A  B 

furnishes  a  simple  construction  for  the  ellipse  when  the  lengths 
and  the  position  of  a  pair  of  conjugate  diameters  are  known. 

The  circle  is  a  special  case  of  an  ellipse  in  which  the  semi-diame- 
ters are  of  equal  length.  The  equation  of  the  circle,  referred  to  a 
pair  of  diameters  as  coordinate  axes,  is  therefore 

x2  +  y2  =  a2. 


EQUATIONS  OF  CURVES  103 

3.  The  Pfii'ubola.— Choose  any  diameter  as  OX  for  the  X-axis; 
and  the  tangent,  OY,  at  its  extremity  for  the  F-axis  (Fig.  65). 
Let  P(x,  y)  and  Q  (xi,  iji)  be  any  two  points  on  the  curve,  and  R  the 
infinitely  distant  point  on  OX.  The  quadrangle  QOPR  is  then 
inscribed  in  the  curve.  The  tangents  at  0  and  R  must  meet  on  the 
line  ST  (Art.  57).  Thus  ST  is  parallel  to  OY.  Also  SP  and  TQ 
are  parallel  lines.  Hence, 

l_^_Lr_^_ll 

x  ~  OD  ~  LO  ~  LO  ~  LO 

and 

Vl  -  EQ  _  LS  _  DP  _    y_ 

xi  ~  OE  ~  LO  ~  LO  ~  LO  {  ' 

Eliminating  LO  from  (1)  and  (2)  we  have 

f/2  _  l/i2. 

X  Xi 

The  ratio  y2/x  is,  therefore,  the  same  for  all  points  on  the  parabola. 
Hence:  The  equation  of  a  parabola,  referred  to  any  diameter  and  the 
tangent  at  its  extremity  as  coordinate  axes,  is 

yz  —  kx, 
where  k  is  a  constant. 

The  parabola  is  completely  determined  by  three  fixed  points 
Q,  0,  R  and  the  tangents  at  0  and  R;  and  is  generated  by  project- 
ively  related  sheaves  of  rays  with  centers  at  0  and  R,  correspond- 
ing rays  meeting  in  the  various  positions  of  P.  This  remark 
furnishes  an  easy  construction  for  the  parabola. 

The  results  of  this  article  establish  completely  the  identity  of  the 
curves  of  second  order  with  the  conies  of  analytic  geometry. 

Exercises 

1.  Construct  an  ellipse  according  to  the  suggestions  made  in  2  of 
the  preceding  article. 

2.  Construct  a  parabola  according  to  the  suggestions  in  3  of  the 
preceding  article. 

3.  Let  u  and  u\  be  a  pair  of  parallel  lines  and  AB  any  transversal 
cutting  u  in  A  and  u\  in  B;  let  TS  be  a  second  transversal,  inclined 
to  the  first,  and  meeting  u  in  T  and  u\  in  S.     If  TS  moves  parallel 
to  itself,  it  cuts  u  and  u\  in  point-rows  in  perspective  position.     Show 
that  the  sheaves  of  rays  described  by  AS  and  by  BT  are  projectively 


104  PROJECTIVE  GEOMETRY  [§79 

related  and  generate  a  hyperbola  of  which  AB  is  a  diameter;  that  the 
conjugate  diameter  is  parallel  to  TS;  and,  if  AB  =  2a  and  TS  =  26, 
the  equation  of  the  hyperbola,  referred  to  these  diameters  as  coordi- 
nate axes,  is  z2/a2  —  t/2/62  =  1,  AB  being  the  X-axis. 

79.  Diametral  Planes  and  Axes  of  Cylinders. — The  properties 
of  conies  which  depend  upon  the  Pascal  theorem  or  the  Brianchon 
theorem  have  been,  thus  far,  transferable  to  the  cone  by  simple 
projection.  For  example,  we  have  seen  (Art.  72)  that  the  theory 
of  poles  and  polar  lines  with  respect  to  a  given  conic  can  be  trans- 
ferred at  once  by  projection  into  a  like  polar  theory  with  respect  to 
a  given  cone  in  the  bundle  of  rays  whose  center  is  the  vertex  of 
the  cone. 

The  properties  of  diameters  and  axes  are  only  special  cases  of 
the  properties  of  poles  and  polar  lines  with  respect  to  a  fixed  conic. 
But  when  we  attempt  to  develop  corresponding  properties  for  the 
cone  by  means  of  projection,  a  difficulty  at  once  arises  because  the 
ideal  plane  does  not  pass  through  the  vertex  of  the  cone  unless  the 
vertex  is  infinitely  distant;  that  is,  unless  the  cone  is  a  cylinder. 
For  the  cylinder,  then,  we  have  a  theory  of  diametral  planes  and 
axes  derivable  by  projection  from  the  properties  of  the  diameters 
and  axes  of  the  conic  of  which  the  cylinder  is  a  projector.  Thus, 
the  pole-ray  of  the  ideal  plane  is  the  axis  of  the  cylinder.  The  axis 
is  the  projector  of  the  center  of  the  conic.  The  axis  of  a  parabolic 
cylinder  is  an  ideal  line;  the  axis  of  an  elliptic  cylinder  or  of  an 
hyperbolic  cylinder,  is  an  actual  line. 

Any  plane  through  the  axis  of  a  cylinder  is  a  diametral  plane. 
The  pole-ray  of  a  diametral  plane  is  an  ideal  line.  A  diametral 
plane  is  the  projector  of  a  diameter  of  the  conic  of  which  the 
cylinder  is  a  projector.  Two  diametral  planes,  each  containing 
the  pole-ray  of  the  other,  are  conjugate  diametral  planes. 

Any  plane  perpendicular  to  the  axis  of  an  elliptic  cylinder,  or  an 
hyperbolic  cylinder,  cuts  the  cylinder  in  a  conic.  The  axes  of  this 
conic,  together  with  the  axis  of  the  cylinder,  determine  a  pair  of 
diametral  planes  perpendicular  to  each  other.  These  planes  are 
the  principal  planes  of  the  cylinder.  Any  plane  parallel  to  one  of 
the  principal  planes  cuts  the  cylinder  in  a  pair  of  lines  equidistant 
from  the  other  principal  plane.  The  principal  planes  are, 
therefore,  planes  of  symmetry. 


CHAPTER  X 
RULED  SURFACES  OF  SECOND  ORDER 

80.  Ruled  Surfaces. — When  a  straight  line  moves  continu- 
ously in  space,  it  describes,  or  generates,  a  ruled  surface.  The 
form  of  the  surface  depends  upon  the  law  governing  the  motion 
of  the  line.  Thus,  if  the  line  passes  through  a  fixed  point  and  meets 
a  fixed  curve,  it  describes,  or  generates,  a  cone.  If  the  fixed 
point  is  ideal,  the  cone  generated  is  a  cylinder;  if  the  fixed  curve 
is  a  straight  line,  the  surface  generated  is  a  plane. 

When  a  point  describes  continuously  one  of  two  projectively 
related  point-rows  which  generate  a  regulus,  its  corresponding 
point  describes  continuously  the  other  point-row  (Art.  39,  theorem 
VI),  and  the  line  joining  corresponding  points  describes  continu- 
ously the  regulus.  The  rays  of  a  regulus  thus  lie  on  a  smooth  sur- 
face which  is  called  a  ruled  surface  of  second  order.  Each  ray 
of  the  regulus  is  called  a  generator  of  the  surface. 

The  following  theorem  is  fundamental  for  the  ruled  surfaces 
of  second  order. 

Theorem  XII. — A  surface  of  second  order  upon  which  lies  a 
regulus  V  contains  a  second  regulus  U.  Each  ray  of  either  regulus 
meets  all  the  rays  of  the  other. 

Let  u  and  u\  be  the  two  projectively  related  point-rows  which 

generate  the  regulus  V  (Fig.  66),  the  lines  v,  v\,  Vz,  v$, being 

rays  of  V.  A  sheaf  of  planes  having  any  one  of  these  rays  as  an 
axis  cuts  out  projectively  related  point-rows  along  any  two  of  the 
other  rays.  For  example,  the  sheaf  of  planes  whose  axis  is  v\  cuts 
out  projectively  related  point-rows  along  v  and  Vz.  The  lines 
joining  corresponding  points  on  v  and  Vz  constitute  rays  of  a  second 
regulus  U  to  which  belong  the  lines  u  and  u\.  Every  ray  of  U 
meets  the  three  rays  v,  v\,  v2  and,  in  consequence,  must  meet 
all  the  rays  of  V.  For  the  two  sheaves  of  planes  determined  by 
Uz  and  the  two  point-rows  u  and  u\  are  coaxial  and  projectively 

105 


106 


PROJECTIVE  GEOMETRY 


I  §81 


related;  and  the  three  planes  uzA,  uzB,  and  u2C  coincide  re- 
spectively with  their  corresponding  planes  UzA\,  uiB\,  uiC\.  The 
two  sheaves  thus  have  three  self-corresponding  planes  and  conse- 
quently coincide  throughout  (theorem  VII).  Since  the  plane 
UzD  coincides  with  the  plane  u2Di,  Uz  meets  Vz]  and  similarly 
must  meet  the  other  rays  of  F.  In  the  same  way  we  can  show 

that  the  rays  u3,  ut, must  meet  all  the  rays  of  V.     The  surface 

that  contains  the  regulus  V,  therefore  also  contains  the  regulus  U. 
No  two  rays  of  either  regulus  ever  meet  each  other,  for  then 
all  the  rays  of  that  regulus  would  be  coplanar;  but  any  ray  of 
either  regulus  meets  all  the  rays  of  the  other. 


FIG.  66. 

Through  any  point  on  a  ruled  surface  of  second  order  there 
pass  two  rays,  one  from  each  regulus  lying  on  the  surface. 

A  straight  line,  not  lying  wholly  on  the  surface,  cannot  meet 
the  surface  in  more  than  two  points.  For,  if  a  line  has  three 
points  in  common  with  the  surface,  it  meets  three  rays  of  one 
regulus,  and  consequently  all  the  rays  of  that  regulus,  and  so 
must  belong  to  the  other  regulus.  It  is  for  this  reason  that  the 
surface  is  said  to  be  of  second  order. 

81.  Sections  of  a  Surface  of  Second  Order. — A  surface  of 
second  order  is  cut  by  any  plane,  not  containing  a  ray  of  either 
regulus,  in  a  conic. 


RULED  SURFACES  OF  SECOND  ORDER 


107 


For  the  cutting  plane  meets  either  regulus  in  a  conic  (Art.  47). 

If  the  cutting  plane  contains  a  ray  of  either  regulus,  we  have 
the  following  theorem: 

Any  'plane  which  passes  through  a  ray  of  either  regulus  contains 
a  ray  of  the  other  regulus  and  does  not  meet  the  surface  outside  these 
rays. 

For,  suppose  the  plane  a  passes  through  the  ray  Vi  of  the  regulus 
V  and  meets  two  other  rays  of  the  same  regulus  in  the  points 
M  and  N  (Fig.  67).  The  straight  line  MN  meets  three  rays 
of  the  regulus  V  and  must,  therefore,  meet  all  of  them.  It  is 
then  a  ray  of  the  other  regulus  U,  say  the  ray  u,-.  Hence  a  has 
the  two  lines  Vi  and  u,-  in  common  with  the  surface.  That  it 
can  have  no  point  outside  these  lines  in  common  with  the  surface 


/"  /- 


FIG.  67. 

follows  in  this  way ;  suppose  there  is  a  point  P,  not  lying  on  either 
Vi  or  u,-,  but  which  is  common  to  the  plane  and  to  the  surface. 
We  can  then  draw  any  number  of  lines  through  P  meeting  both 
Vi  and  Uj  and  therefore,  having  three  points  in  common  with  the 
surface.  But  this  is  impossible  unless  all  these  lines  lie  entirely 
on  the  surface  (Art.  80),  in  which  case  the  entire  plane  must  form 
a  part  of  the  surface.  But  a  plane  cannot  form  part  of  the  surface 
since  no  two  rays  of  either  regulus  lie  in  any  plane.  We  conclude, 
therefore,  that  a  cannot  meet  the  surface  outside  the  two  lines 
Vi  and  Uj.  The  plane  a  can  be  any  plane  through  any  ray  of  either 
regulus. 

82.  Tangent    Lines    and    Tangent    Planes. — A    'plane    which 
contains  a  ray  of  either  regulus  on  a  ruled  surface  of  second  order 


108  PROJECTIVE  GEOMETRY  [§83 

is  a  tangent  plane  to  the  surface,  the  point  of  contact  being  the  inter- 
section of  the  two  rays  lying  in  the  plane. 

For  all  the  lines  which  pass  through  the  intersection  of  the 
two  rays  and  lie  in  the  plane  cannot  meet  the  surface  outside  this 
point  of  intersection.  These  lines  are,  therefore,  tangent  lines  to 
the  surface  at  the  point  of  intersection;  and  the  plane  containing 
them  is  the  tangent  plane  to  the  surface  at  that  point. 

83.  Tangent  Cones. — If  we  project  a  ruled  surface  from  a 
point  S  not  lying  on  the  surface,  we  obtain  a  sheaf  of  planes  of 
second  class.     For  the  projector  of  a  regulus,  from  any  point 
not  on  a  ray  of  the  regulus,  is  a  sheaf  of  planes  of  second  class 
(Art.  47).     Each  plane  of  the   sheaf  contains  two   generators 
of  the  surface,  one  from  each  regulus,  and  is  therefore,  tangent 
to  the  surface  at  the  point  of  intersection  of  the  generators. 
The  sheaf  of  planes  envelopes  a  cone  whose  center  is  S.    The 
cone  is  the  tangent  cone  to  the  surface  from  S.    The  line  joining 
S  to  the  point  of  contact  on  any  plane  of  the  sheaf  of  planes  is  a 
ray  of  the  tangent  cone.     The  lines  drawn  from  the  eye  touching 
the  surface  are  rays  of  the  tangent  cone  whose  vertex  is  the  eye. 

84.  Polar  Planes. — The  points  of  contact  on  the  rays  of  a  tangent 
cone  lie  in  one  and  the  same  plane. 

No  three  of  the  points  of  contact  can  lie  on  the  same  straight 
line,  for  then  this  line  must  lie  entirely  on  the  surface  and  the 
tangent  planes  at  the  three  points  must  coincide  with  the  plane 
determined  by  this  line  and  the  vertex  of  the  cone;  and  the  rays 
of  one  regulus  which  pass  through  the  three  points  must  lie  in  the 
plane  so  determined.  But  this  is  impossible.  Hence,  three  of 
the  points  of  contact  determine  a  plane  which  cuts  the  surface 
in  a  conic  and  also  cuts  the  tangent  cone  in  a  conic.  But  these 
two  conies  coincide  throughout,  since  they  have  three  points  and 
the  tangents  at  them  in  common  (Art.  46). 

If  A  is  the  vertex  of  a  tangent  cone  and  a  is  the  plane  contain- 
ing the  points  of  contact  on  the  rays  of  the  cone,  a  is  called  the 
polar  plane  of  A  with  respect  to  the  surface. 

The  surface-points  on  any  line  through  A  are  separated  harmonic- 
ally by  A  and  the  polar  plane  of  A . 

For  any  plane  through  A,  which  is  not  a  tangent  plane,  meets 
the  surface  in  a  conic,  and  the  polar  plane  of  A  in  the  chord  of 
contact  of  tangents  to  the  conic  from  A. 


§87]  RULED  SURFACES  OF  SECOND  ORDER  109 

86.  Circumscribing  Tetrahedrons. — Let  P  and  PI  be  any  two 
points  on  a  surface  of  second  order;  a  and  61,  the  two  generators 
passing  through  P;  and  ai  and  6,  the  two  generators  through  PI 
(Fig.  68).  Since  each  ray  of  either  regulus  meets  all  the  rays 
of  the  other,  it  follows  that  a  meets  b  in  a  point  Q  and  ai  meets 
61  in  a  point  Qi.  Each  face  of  the  tetrahedron PP\QQ\ is  a  tangent 
plane  to  the  surface,  the  point  of  contact  being  a  vertex.  The 
tetrahedron  thus  formed  circumscribes  the  surface  and  is  com- 
pletely determined  by  the  pair  of  points  P,  P\. 

86.  The  Class  of  a  Ruled  Surface  of  Second  Order.— The 
class  of  a  surface  is  defined  as  the  maximum  number  of  tangent 
planes  that  can  be  drawn  through  a 

straight  line  not  lying  entirely  on  the 
surface. 

A  ruled  surface  of  second  order  is  of 
second  class. 

For  a  line,  not  lying  entirely  on  the 
surface,  cannot  meet  the  surface  in 
more  than  two  points;  and  but  two 
of  the  faces  of  the  circumscribing 
tetrahedron,  determined  by  these 
points,  contain  the  line. 

If  a  line  touches  the  surface,  but 
one  tangent  plane  can  be  drawn  jr 
through  the  line,  namely,  the  plane 
determined  by  the  generators  of  the 
surface  through  the  point  of  contact  of  the  line. 

If  a  line  does  not  meet  the  surface,  no  tangent  plane  can  be 
drawn  through  the  line;  for  the  line  does  not  then  meet  any  of 
the  generators  of  the  surface. 

If  a  line  lies  on  the  surface,  every  plane  through  it  is  tangent 
to  the  surface. 

We  conclude,  therefore:  As  many  tangent  planes  to  a  surface 
of  second  order  can  be  drawn  through  a  given  line  as  the  line  has 
points  in  common  with  the  surface. 

87.  Classification  of  Ruled  Surfaces  of  Second  Order. — We 
have  seen  that  a  plane  may  meet  a  ruled  surface  of  second  order 
in  a  conic  or  in  two  straight  lines,  and  in  the  latter  case,  that  the 


110 


PROJECTIVE  GEOMETRY 


[§87 


plane  is  tangent  to  the  surface  (Art.  81).  If  the  ideal  plane  meets 
the  surface  in  a  conic,  the  surface  is  called  a  hyperboloid  of  one 
sheet  or  a  unapartite  hyperboloid  (Fig.  69).  If  the  ideal  plane 
is  tangent  to  the  surface,  that  is,  if  it  contains  a  ray  of  each  regulus 
the  surface  is  called  a  hyperbolic  paraboloid  (Fig.  70). 

The  hyperboloid  of  one  sheet  is  cut  by  an  arbitrary  plane  in 
an  ellipse,  a  parabola,  or  a  hyperbola,  according  as  the  cutting 


FIG.  69. 

plane  meets  the  infinitely  distant  conic  in  no  point,  in  one  point, 
or  in  two  points. 

The  hyperbolic  paraboloid  is  cut  by  an  arbitrary  plane  in  a 
hyperbola,  or  a  parabola,  according  as  the  cutting  plane  meets  the 
infinitely  distant  generators  in  two  distinct  points,  or  passes 
through  their  point  of  intersection. 

The  tangent  cone  to  a  hyperboloid  of  one  sheet,  whose  points  of 
contact  lie  on  the  infinitely  distant  conic,  is  the  asymptotic  cone. 
The  vertex  of  the  asymptotic  cone  is  the  center  of  the  surface. 
The  ideal  plane  is  the  polar  plane  of  the  center. 


§87]  RULED  SURFACES  OF  SECOND  ORDER 


Exercises 


111 


1.  Show  that  the  locus  of  the  vertex  of  a  cone  of  second  order,  to 
which  the  six  sides  of  a  twisted  hexagon  are  tangent,  is  a  ruled  surface 
of  second  order  of  which  the  three  principal  diagonals  of  the  hexagon 
are  generators. 

2.  Show  that  the  three  principal  diagonals  of  a  twisted  hexagon, 
whose  six  sides  lie  upon  a  ruled  surface  of  second  order,  intersect  in 
one  point. 

3.  A  point-row  u  and  a  sheaf  of  rays  S  are  projectively  related  but 
do  not  lie  in  parallel  planes.     Show  that  lines  drawn  through  the 
points  of  u  parallel  to  the  corresponding  rays  of  S  constitute  one 
regulus  of  a  hyperbolic  paraboloid. 


FIG.  70. 


4.  A  point-row  u  is  projectively  related  to  a  sheaf  of  planes  v,  and 
u  is  not  perpendicular  to  the  axis  of  v.  Show  that  lines  drawn  through 
the  points  of  u  perpendicular  to  the  corresponding  planes  of  v  form  one 
regulus  of  a  hyperbolic  paraboloid. 

6.  Perpendiculars  to  a  surface  of  second  order  are  erected  at  the 
points  of  a  line  lying  upon  the  surface.  Show  that  these  perpendic- 
ulars form  one  regulus  of  a  hyperbolic  paraboloid.  Show  also  that 
the  generators  of  the  paraboloid  are  parallel  to  a  pair  of  perpendicular 
planes.  Such  a  paraboloid  is  said  to  be  equilateral. 

6.  Show  that  lines  drawn  through  any  point  parallel  to  the  rays  of  a 
regulus  lie  in  one  plane  or  upon  a  cone,  according  as  the  regulus  belongs 
to_a  hyperbolic  paraboloid  or  to  a  hyperboloid  of  one  sheet. 


CHAPTER  XI 
PROTECTIVELY  RELATED  ELEMENTARY  FORMS 

88.  Four  Harmonic  Elements  of  an  Elementary  Form. — An 
elementary  form  is  generated  by  two  projectively  related  primi- 
tive forms  and  any  four  harmonic  elements  of  either  of  these 
primitive  forms  determine,  with  their  corresponding  elements, 
four  harmonic  elements  of  the  elementary  form.  For  example, 
four  points  of  a  conic  are  harmonic  whenever  they  are  the  inter- 
sections of  corresponding  rays  in  two  harmonic  pencils  belonging 
to  the  sheaves  of  rays  which  generate  the  conic.  Since  a  conic 

is  projected  from  any  two  of  its  points 
in  projectively  related  sheaves  of  rays 
(Art.  45,  3),  four  points  of  a  conic  are 
harmonic  whenever  they  are  projected 
from  any  other  point  of  the  curve  in 
a  harmonic  pencil  of  rays. 

If  a  conic  is  cut  by  two  conjugate  lines 
AC  and  BD  (Fig.   71),   the  points  of 
intersection  A,B,C,D  are  four  harmonic 
•p       71  points  of  the  conic. 

For,  if  Q  is  the  pole  oi  AC  and  R  is 

the  pole  of  BD,  then  QBPD   and  APCR  are  harmonic  ranges 
(Art.  64).     Hence  we  have, 

A(QBPD — )  7\  D(APCR — ); 

corresponding  rays  intersecting  in  the  points  of  the  conic  A,  B, 

C,  D, .  Therefore,  the  points  A,  B,  C,  D  are  harmonic  by 

definition. 

Conversely:  //  the  four  points  A,B,C,Dofa  conic  are  harmonic 
points,  AC  and  BD  are  conjugate  lines  with  respect  to  the  conic. 

For  the  pencil  formed  by  the  lines  DA,  DB,  DC,  and  the  tan- 
gent at  D  is  harmonic;  and  the  line  AC  cuts  this  pencil  in  the 

112 


§89]  ELEMENTARY  FORMS  113 

harmonic  range  APCR.  Hence  AC  passes  through  the  pole  of 
BD. 

In  general,  four  elements  of  an  elementary  form  are  harmonic 
whenever  they  are  projected  from,  or  cut  by,  any  fifth  element 
of  the  form  in  four  harmonic  elements  of  a  primitive  form. 

Thus :  The  tangents  at  four  harmonic  points  of  a  conic  form  four 
harmonic  rays  of  the  envelope  of  second  class  whose  rays  are  tangents 
to  the  conic. 

For  the  pencils  Q(APCR)  and  R(QBPD)  (Fig.  71)  are  harmonic 
and  cut  the  tangents  at  B  and  C,  respectively,  in  harmonic  ranges. 
Pairs  of  corresponding  points  on  these  ranges  are  joined  by  the 
tangents  at  A,  B,  C,  D. 

Four  rays  of  a  cone  are  harmonic  if  they  are  projected  from  any 
fifth  ray  of  the  cone  in  a  harmonic  pencil  of  planes;  four  planes 
of  a  sheaf  of  planes  of  second  class  are  harmonic  if  they  are  cut 
by  any  fifth  plane  of  the  sheaf  in  a  harmonic  pencil  of  rays. 

The  above  general  definition  does  not  apply  to  a  regulus, 
since  four  rays  of  a  regulus  are  never  cut  by  any  fifth  ray  of  the 
same  regulus.  Four  rays  of  a  regulus  are  harmonic  whenever 
they  are  cut  in  a  harmonic  range  by  any  ray  of  the  other  regulus 
on  the  same  surface  of  second  order.  If  four  rays  of  a  regulus 
are  cut  in  a  harmonic  range  by  any  ray  of  the  other  regulus,  they 
are  cut  in  harmonic  ranges  by  all  rays  of  the  other  regulus. 

The  definition  of  four  harmonic  elements  of  a  primitive  form 
(Art.  21)  is  thus  extended  to  include  the  elementary  forms. 

89.  Elementary  Forms  in  Perspective  Positions  with  Primitive 
Forms. — An  elementary  form  is  in  perspective  position  with 
either  of  two  primitive  forms  which  generate  it.  A  conic  is 
in  perspective  position  with  a  sheaf  of  rays  whose  center  is  any 
point  of  the  conic,  and  whose  rays  pass  through  their  corre- 
sponding points  of  the  conic;  an  envelope  of  second  class  is  in 
perspective  position  with  the  point-row  it  cuts  upon  any  one  of 
its  rays;  a  cone  is  in  perspective  position  with  a  sheaf  of  planes 
whose  axis  is  any  one  of  it's  rays,  the  planes  of  the  sheaf  containing 
their  corresponding  rays  of  the  cone;  a  regulus  is  in  perspective 
position  with  the  point-row  it  cuts  upon  any  ray  of  the  other 
regulus  on  the  same  surface  of  second  order.  The  primitive 
form,  in  these  examples,  may  be  regarded  as  one  of  two  pro- 

8 


114  PROJECTIVE  GEOMETRY  [§90 

jectively  related  primitive  forms  which  generate  the  elementary 
form. 

90.  Elementary  Forms  in  Perspective  Positions  with  Each 
Other. — Two    correlated   elementary  forms,    consisting   of   like 
elements,  are  in  perspective  position  with  each  other,  if  each  is  a 
section  of,   or  projector  of,   the  same  third  elementary  form. 
Thus,  two   conies   are  in  perspective  position  with  each  other 
if  each  is  a  section  of  the  same  cone,  or  of  the  same  regulus. 

Two  correlated  elementary  forms,  consisting  of  unlike  elements, 
are  in  perspective  position  with  each  other,  if  each  element  of 
one  lies  on,  or  passes  through,  the  corresponding  element  of  the 
the  other.  Thus,  a  cone  and  a  conic  are  in  perspective  position, 
if  each  point  of  the  conic  lies  on  the  corresponding  ray  of  the 
cone ;  a  regulus  is  in  perspective  position  with  a  sheaf  of  planes  of 
second  class,  if  each  ray  of  the  regulus  lies  in  the  corresponding 
plane  of  the  sheaf. 

The  concept  of  perspective  position,  defined  for  primitive  forms 
(Art.  15),  is  thus  extended  to  include  the  elementary  forms. 

We  shall  often  speak  of  the  primitive  forms  and  the  elementary 
forms  together  as  fonns;  and  we  may  obviously  consider  chains 
of  perspectivity  (Art.  34)  which  may  include  elementary  forms 
as  well  as  primitive  forms. 

Two  forms  in  a  chain  of  perspectivity  are  correlated  so  that 
to  any  four  harmonic  elements  of  either  always  correspond  four 
harmonic  elements  of  the  other. 

91.  Projectively  Related  Forms. — We  are  now  in  a  position 
to    extend    the  definition    of    projective  relationship   (Art.   35) 
so   as   to   include  the  elementary  forms.     Thus,  two  correlated 
forms  are  projectively  related  if  any  four  harmonic  elements 
of  either  correspond  to  four  harmonic  elements  of  the  other.     For 
example,  two  forms  are  projectively  related  if  they  are  in  perspec- 
tive position,  or  if  they  are  contained  in  a  chain  of  perspectivity. 

Two  projectively  related  forms  can  always  be  connected  by  a  chain 
of  perspectivity. 

Consider,  for  example,  two  projectively  related  conies  k  and 
ki.  Let  £  be  a  sheaf  of  rays  in  perspective  position  with  k  and 
Si  a  sheaf  of  rays  in  perspective  position  with  ki.  S  and  Si 


§92]  ELEMENTARY  FORMS  115 

can  be  connected  by  a  chain  of  perspectivity  as  in  Art.  43.     We 
then  have  the  chain  of  perspectivity 

k7\  S  7\  $2  X£i  7\  ki. 

When  two  forms  are  protectively  related,  we  shall  say  that 
a  projectivity  exists  between  them;  and  we  shall  often  think  of 
this  projectivity  as  transforming  either  form  into  the  other, 
the  transformation  being  effected  by  a  chain  of  perspectivity 
connecting  the  two  forms. 

92.  Determination  of  Proj active  Relationship. — Theorem 
VIII,  Art.  43,  can  now  be  extended  to  include  the  elementary 
forms.  Thus:  A  projectivity  can  always  be  established  between  any 


FIG.  72. 

two  forms  so  that  any  three  elements  of  the  one  shall  correspond 
to  three  elements  of  the  other  chosen  arbitrarily. 

Thus,  for  example,  to  establish  a  projectivity  between  the 
two  conies  k  and  ki  (Fig.  72)  so  that  the  points  A,  B,  C  of  k  shall 
correspond  respectively  to  the  points  Ai,  BI,  Ci,  of  ki,  join  A  and 
A  i  and  let  the  line  AAi  meet  k  again  in  S,  and  ki  again  in  T. 
Project  k  from  S  and  k\  from  T,  and  thus  obtain  two  sheaves  of 
rays,  S  and  T,  which  are  projectively  related  and  in  perspective 
position.  Hence  the  chain  of  perspectivity 


If  D  is  any  point  of  k,  the  corresponding  point,  DI  of  k\,  can  be 
immediately  constructed. 

In  a  similar  manner,  a  chain  of  perspectivity  can  be  constructed 


116  PROTECTIVE  GEOMETRY  [§93 

between  any  two  protectively  related  forms,  having  given  three 
pairs  of  corresponding  elements. 

Two  projectively  related  elementary  forms  may  be  used  to 
generate  a  new  form,  provided  a  pair  of  corresponding  elements 
determines  a  third  element  variable  with  the  pair.  Thus,  the 
projectively  related  conies  just  considered  generate  a  sheaf 
of  rays  of  higher  class. 

93.  Superposition  of  Projectively  Related  Forms. — Two  forms, 
consisting  of  like  elements,  may  be  superposed  as  in  Art.  40. 
The  two  forms  then  have  a  common  base,  or  support.     Thus, 
two  point-rows  of  second  order  are  superposed  when  they  lie 
on  the  same  conic;  two  envelopes  of   second  class   are  super- 
posed when  they  each  form  the  system  of  tangents  to  the  same 
conic;  two  reguli  are  superposed  when  they  each  belong  to  the 
same  regulus;  and  so  on. 

When  two  projectively  related  forms  are  superposed,  we  shall 
say  that  they  constitute  a  projectivity  upon  their  common  base. 
Thus,  two  projectively  related  and  superposed  point-rows  of 
second  order  constitute  a  projectivity  upon  their  common  conic; 
two  projectively  related  and  superposed  point-rows  of  first  order 
form  a  projectivity  upon  their  common  support:  and  so  on. 

A  projectivity  may  be  established  upon  any  form  by  choos- 
ing arbitrarily  three  pairs  of  corresponding  elements. 

We  shall  often  think  of  a  projectivity  upon  a  form  as  a  trans- 
formation which  changes  the  form  into  itself;  the  transformation 
being  affected  by  a  chain  of  perspectivity  connecting  the  two  forms 
which  constitute  the  projectivity. 

94.  Double  Elements  of  a  Projectivity. — An   element   which 
corresponds  to  itself  in  a  projectivity  upon  any  form  is  called  a 
self-corresponding  element,  or  a  double  element,  of  the  pro- 
jectivity. 

Von  Staudt's  fundamental  theorem  (Art.  41)  can  be  extended 
to  include  the  elementary  forms,  and  may  be  restated  as  follows : 

A  projectivity  upon  any  form  cannot  have  more  than  two  double 
elements,  unless  every  element  is  a  double  element. 

For,  if  there  are  three  double  elements,  the  two  forms  which 
constitute  the  projectivity  can  be  projected  from,  or  cut  by, 
the  same  fourth  element  of  their  common  base  in  two  projectively 


ELEMENTARY  FORMS 


117 


related  primitive  forms  having  three  self-corresponding  elements 
and  consequently  coinciding  throughout.  The  forms  themselves, 
therefore,  coincide  throughout,  each  element  being  a  double 
element  of  the  projectivity.  Thus,  if  a  projectivity  on  a  conic 
has  three  double  ppints,  the  two  point-rows  of  second  order  which 
constitute  the  projectivity  can  be  projected  from  the  same  fourth 
point  of  their  common  conic  in  two  projectively  related  sheaves 
of  rays  having  three  self-corresponding  rays  and,  therefore,  coin- 
ciding throughout.  Each  point  of  the  conic  is  then  a  double  point 
of  the  projectivity. 

95.  The  Axis  of  a  Projectivity  on  a  Conic. — In  a  projectivity 
on  a  conic  (Fig.  73),  let  the  points  A,  B,  C  correspond  respectively 
to  the  points  Ai,  B\,  Ci,  so  that 

ABC A  A  iBiCi . 


From  A  project  A 
A  i  project  ABC 


i  --  ,  and  from 
Wethusobtain 
two  sheaves  of  rays  in  perspective  posi- 
tion, since  the  ray  AAi  corresponds 
to  itself.  Corresponding  rays  in  these 
sheaves  meet  upon  a  straight  line  u, 
called  the  axis  of  the  projectivity  upon 
the  conic.  The  chain  of  perspectivity 
ABC  -  = 


FIG.  73. 


enables  us  to  con- 


struct as  many  pairs  of  corresponding  points  as  may  be  desired. 
In  particular,  the  points  M  and  N,  in  which  the  axis  meets  the 
conic,  are  the  double  points  of  the  projectivity.  Hence: 

A  projectivity  on  a  conic  has  two  double  points,  one  double  point, 
or  no  double  points,  according  as  the  axis  of  the  projectivity  meets 
the  conic  in  two  points,  one  point,  or  no  points. 

If  we  consider  the  inscribed  hexagon  AB\CA\BC\t  we  see  that 
the  lines  BCi  and  B:C  also  meet  on  the  axis  u.  We  should  then 
have  obtained  the  same  axis  had  we  chosen  B  and  BI,  or  C  and  Ci 
as  centers  from  which  to  project  the  two  point-rows  AiBiCi  —  •  — 
and  ABC  -  .  In  other  words,  the  axis  is  the  Pascal  line 
(Art.  53)  belonging  to  the  hexagon  ABiCAiBC\. 

We  can  construct  the  axis  of  a  projectivity  on  a  conic  by 


118 


PROJECTIVE  GEOMETRY 


§96 


joining  pairs  of  points  "crosswise."  Thus,  the  pairs  of  lines 
ABi,  AiB;  AC  i,  A\C;  BC\,  BiC  meet  on  the  axis.  These  pairs  of 
lines  can  be  most  easily  recognized,  perhaps,  when  the  points  of  one 
point-row  are  placed  over  the  corresponding  points  of  the  other 
point-row,  thus: 

A  B  C— 


96.  The  Center  of  a  Projectivity  on  a  Conic.  —  The  center 
of  a  projectivity  on  a  conic  is  the  pole  of  the  axis  with  respect 
to  the  conic.  Thus,  in  Fig.  73,  U  is  the  center  of  the  projectivity 
determined  by  the  point-rows  ABC  -  and  AiB\C\  —  -  where 
U  is  the  pole  of  the  axis  u  with  respect  to  the  conic. 

Since  the  tangents  at  A,  B,  C,  -  form  an  envelope  of  second 
class  projectively  related  to  the  point-row  ABC  -  (Art.  88),  it 


N 


FIG.  74. 


follows  that  the  two  envelopes,  formed  by  the  tangents  at  A,  B,  C, 

and  the  tangents  at  A\,  Bi,  C\, ,  are  projectively  related  to 

each  other  and  thus  constitute  a  projectivity  on  the  envelope  formed 
by  all  the  tangents  to  the  conic.  The  tangents  at  the  pairs  of  corre- 
sponding points,  A,  AI,  B,  Bi  and  C,  C\,  are  opposite  sides  of  a 
circumscribed  hexagon  whose  Brianchon  point  (Art.  53)  is  U  (cf. 
also  Art.  70,  Fig.  54). 

A  projectivity  on  a  conic  has  two  double  points,  one  double  point, 
or  no  double  points,  according  as  the  center  of  the  projectivity  lies 
outside  the  conic,  on  the  conic,  or  'inside  the  conic. 


ELEMENTARY  FORMS 


119 


97.  Double  Elements  of  a  Projectivity  on  Any  Form. — The 

problem  of  constructing  the  double  elements  of  a  projectivity  on 
any  form  can  be  reduced  by  projection  or  section  to  the  problem 
of  constructing  the  double  points  of  a  projectivity. on  a  conic.  For 
example,  suppose  a  projectivity  on  a  sheaf  of  rays  of  first  class  is 
determined  by  the  three  pairs  of  corresponding  rays  a,  a\\  b,  61 ; 
and  c,  Ci  (Fig.  74).  Cut  the  sheaf  by  a  conic  passing  through  the 
center  S  (for  convenience,  a  circle),  and  thus  obtain  a  projectivity 
upon  the  conic.  The  double  points,  M  and  N,  of  the  projec- 
tivity on  the  conic  determine  the  double  rays,  m  and  n,  of  the 
projectivity  on  the  sheaf  S. 

A.  projectivity  on  a  point-row  of  the  first  order  can  be  projected 
from  any  point  S  in  a  projectivity  on  the  sheaf  whose  center  is  S. 
The  construction  of  the  double  points  of  the  given  projectivity 
then  follows  as  above.  This  is  illustrated  in  Fig.  74. 

A  projectivity  on  an  envelope  of  second  class  can  be  cut  by  any 
ray  of  the  envelope  in  a  projectivity  on  that  ray.  The  double 
rays  of  the  given  projectivity  can  then  be  constructed  as  above. 

A  projectivity  on  a  cone  can  be  cut  by  a  plane  in  a  prdjectivity 
on  a  conic.  The  double  rays  of  the  given  projectivity  are  deter- 
mined by  the  double  points  of  the  projectivity  on  the  conic. 

98.  Application  of  the  Theorem  in  Art.  95. — We  can  now  solve 
the  following  dual  problems: 


To  construct  the  points  in 
which  a  given  line  u  meets  a  conic 
when  the  conic  is  given  by  five 
conditions;  viz.,  by  five  points,  or 
by  four  points  and  the  tangent  at 
one  of  them,  or  by  three  points 
and  the  tangents  at  two  of  them. 


To  construct  the  tangents  to 
a  conic  from  a  given  point  U 
when  the  conic  is  given  by  five 
conditions;  viz.,  by  five  tangents, 
or  by  four  tangents  and  the  point 
of  contact  on  one  of  them,  or  by 
three  tangents  and  the  points  of 
contact  on  two  of  them. 

On  the  left,  consider  the  sheaves  of  rays  which  generate  the  conic 
(Art.  46)  and  the  projectivity  which  they  determine  upon  the 
given  line  u.  The  points  in  which  the  line  meets  the  conic  are 
the  double  points  of  this  projectivity. 

In  particular,  if  the  given  line  is  the  ideal  line  in  the  plane,  the 
conic  determined  by  the  five  conditions  will  be  an  ellipse,  a  para- 


120  PROJECTIVE  GEOMETRY  [§99 

bola,  or  a  hyperbola,  according  as  the  projectivity  determined 
upon  the  ideal  line  has  no  double  points,  one  double  point,  or  two 
double  points. 

Similarly,  on  the  right,  consider  the  protectively  related  point- 
rows  which  generate  the  envelope  of  tangents  to  the  conic  and  the 
projectivity  which  they  establish  when  projected  from  the  given 
point  U.  The  tangents  from  U  are  the  double  rays  of  this 
projectivity. 

Exercises 

1.  Of  a  projectivity  on  a  conic,  one  double  point  and  two  pairs  of 
corresponding  points  are  known.     Construct  a  number  of  pairs  of 
corresponding  points,  in  particular,  the  other  double  point. 

2.  Of  a  projectivity  on  an  envelope  of  second  class,  one  double  ray 
and  two  pairs  of  corresponding  rays  are  known.     Construct  a  number 
of  pairs  of  corresponding  rays,  in  particular,  the  other  double  ray. 

3.  A  conic  is  given  by  three  tangents  and  the  points  of  contact  on 
two  of  them;  determine  the  point S  of  intersection  of  a  given  line  with 
the  conic. 

4.  A  conic  is  given  by  five  tangents,  no  two  of  which  are  parallel. 
Construct  a  circumscribing  rectangle. 

5.  A  conic  is  tangent  to  the  axes  of  coordinates  at  the  points  (0,  4) 
and  (4,  0)  and  passes  through  the  point  (1,  1).     Show  by  construction 
that  the  conic  is  a  parabola. 

6.  Determine  by  construction  the  nature  of  the  conic  which  passes 
through  the  points  whose  coordinates  are  (—1,  4),  (  —  2,  0),  (0,  3), 
(1,  -2),  and  (2,  3). 

7.  A  cone  is  given  by  five  of  its  rays.     Show  how  to  construct  the 
points  in  which  a  given  line  meets  the  cone. 

8.  A  ruled  surface  of  second  order  is  determined  by  three  rays  of 
regulus.     Show  how  to  construct  the  points  in  which  a  given  line 
meets  the  surface. 

99.  Classification  of  Proj activities  on  a  Form. — A  projectivity 
on  any  form  is  called  hyperbolic,  parabolic,  or  elliptic,  according 
as  it  has  two  double  elements,  one  double  element,  or  no  double 
elements.  Thus,  a  projectivity  on  a  conic  is  hyperbolic,  para- 
bolic, or  elliptic,  according  as  its  axis  meets  the  conic  in  two  dis- 
tinct points,  is  tangent  to  the  conic,  or  does  not  meet  the  conic  at 
all. 


§101]  ELEMENTARY  FORMS  121 

100.  Cyclic  Proj activities. — Suppose  in  a  given  projectivity  on 
any  form,  an  element  E  corresponds  to  EI  and,  in  turn,  EI  cor- 
responds to  E2,  Ez  to  E3,  and  so  on.     We  thus  obtain  a  series  of 
elements 

E,  EI,  EI,  Ez, ,  En, 

of  which  each  corresponds  to  the  next  one  following  it.  Either 
this  series  terminates  so  that  En,  say,  coincides  with  E  (n  being 
a  finite  integer),  or  else  the  series  never  terminates  and  so  contains 
an  infinite  number  of  distinct  elements.  In  the  first  case,  the 
projectivity  is  called  cyclic;  and  in  the  second,  non-cyclic. 
In  the  case  of  a  cyclic  projectivity,  the  series  of  elements 

E,  EI,  Ez,  Ez, ,  En-i  En, 

where  En  coincides  with  E,  is  called  a  cycle ;  the  integer  n  is  the 
order  of  the  cyclic  projectivity. 

101.  Construction  of  Cyclic  Projectivities. — In  what  follows  we 
shall  demonstrate  the  existence  of  cyclic  pro jectivi ties  by  con- 
structing those  of  lower  orders.     Since  a  projectivity  on  any  form 
can  be  reduced  to  a  projectivity  on  a  conic  (Art.  97),  we  may  con- 
fine the  attention  to  projectivities  on  a  conic. 

1.  Cyclic  Projectivities  of  Order  2. — Let  A  and  B  (Fig.  75)  form 
a  cycle;  and  let  any  other  point,  as  C,  correspond  to  D.  The 
projectivity  is  then  determined  by  the  correspondence 

ABC- 

BAD . 

The  pairs  of  lines  BD,  AC  and  AD,  BC  intersect  upon  the  axis  z. 
Any  number  of  pairs  of  corresponding  points  can  be  immediately 
constructed.  In  particular,  the  point  D  corresponds  to  C,  so  that 
D  and  C  form  a  second  cycle  of  the  projectivity.  Again,  the  point 
E  corresponds  to  F,  and  F  corresponds  to  E;  since  the  lines  DE 
and  CF  intersect  on  the  axis.  The  points  E  and  F  thus  form  a 
third  cycle. 
From  the  correspondence 

ABCDEF 

BADCFE , 

we  see  that  the  tangents  at  A  and  B  intersect  on  the  axis  z.     Also, 
the  tangents  at  C  and  D,  and  the  tangents  at  E  and  F,  intersect  on 


122 


PROJECTIVE  GEOMETRY 


[§101 


the   axis.     This   also   follows  from   the   properties  of  inscribed 
quadrangles  (Art.  57). 

Again  (Art.  57),  the  lines  AB,  CD,  EF  pass  through  Z,  the 
center  of  the  projectivity.  Hence  we  have  the  following  impor- 
tant property : 

//  a  projectivity  on  a  conic  is  cyclic  and  of  order  2,  the  lines  joining 
corresponding  points  pass  through  the  center  of  the  projectivity; 
and  the  tangents  at  corresponding  points  meet  on  the  axis  of  the 
projectivity. 

C 


Conversely:  The  rays  of  a  sheaf  of  rays  of  first  class,  whose  center 
is  not  on  a  conic,  cut  the  conic  in  pairs  of  points  of  a  cylic  projectivity 
of  order  2. 

For  a  pair  of  rays  cuts  the  conic  in  the  vertices  of  an  inscribed 
quadrangle.  Opposite  vertices  of  this  quadrangle  are  pairs  of 
points  in  a  cyclic  projectivity  of  order  2.  Any  other  ray  of  the 
sheaf  cuts  the  conic  in  a  pair  of  points  belonging  to  the  same 
projectivity. 

It  follows  that  four  points  A,  B,  C,  D,  chosen  arbitrarily  on  a 
conic,  determine  three  cyclic  projectivities  of  order  2,  namely: 


ABCD- 
BADC- 


ABCD- 
CDAB- 


ABCD- 
DCBA- 


and  consequently 

ABCDABADCKCDABADCBA . 
The  axes  of  these  projectivities  are  the  sides  of  the  diagonal  triangle 


§101] 


ELEMENTARY  FORMS 


123 


determined  by  the  quadrangle  A  BCD  (Fig.  75),  and  their 
centers  are  the  vertices  of  the  diagonal  triangle.  Two  of  the 
projectivities  so  .determined  are  hyperbolic  and  one  is  elliptic. 

In  conclusion  we  can  say: 

A  cyclic  projectivity  of  order  2  on  any  form  is  completely  deter- 
mined by  two  pairs  of  corresponding  elements. 

Also: 

//  1,  2,  3,  4  are  four  elements  in  definite  order,  chosen  arbitrarily 
on  any  form,  then 

1  2  3  4A2  1  4  3A3  4  1  2M  3  2  1. 

2.  Cyclic  Projectivities  of  Order  3. — Any  three  points  A,  B,  C  of 
a  conic  may  be  chosen  for  one  cycle  of  the  projectivity.  The 
projectivity  is  then  entirely  determined  by  the  correspondence 

ABC 

BCA . 


FIG.  76. 

Each  side  of  the  triangle  ABC  meets  the  tangent  at  the  opposite 
vertex  on  the  axis  of  the  projectivity  (Fig.  76).     The  tangents  at 


124 


PROJECTIVE  GEOMETRY 


[§101 


A,  B,  C  form  a  circumscribed  triangle;  and  the  lines  joining  a 
vertex  of  this  triangle  to  the  point  of  contact  on  the  opposite  side 
pass  through  the  center  of  the  projectivity  (cf.  triangle  theorem, 
Art.  63,  exercises  6  and  7). 

Given  any  fourth  point  D,  considered  as  a  point  of  the  point-row 

ABC ,  we  can  immediately  construct  its  corresponding  point 

E  of  the  point-row  BCA .     Similarly,  to  E  of  ABC cor- 


responds F  of  BCA- 


Now  to  F  of  ABC must  correspond 


D  of  BCA .     For,  if  F  does  not  correspond  to  D,  suppose  it 

corresponds  to  D'.     The  correspondence  is  then 


ABCDEF- 
BCAEFD'- 


Hence  the  lines  AD,  CE,  BF,  and  AD'  intersect  in  the  same  point 
on  the  axis.  But  this  cannot  happen  unless  D'  coincides  with  D. 
Therefore,  the  points  D,  E,  F  form  a  second  cycle  of  the  pro- 
jectivity. Any  number  of  cycles  can  be  constructed  in  a  similar 
manner. 

3.  Cyclic  Projectivities  of  Order  4. — Four  points  of  a  conic  cannot 
be  chosen  arbitrarily  to  form  one  cycle  of  a  projectivity,  since  three 
pairs  of  points  completely  determine  a  projectivity.  If,  then,  the 


FIG.  77. 


§101}  ELEMENTARY  FORMS  125 

points  A,  B,  C,  D  form  one  cycle  of  a  projectivity,  the  correspon- 
dence 

ABCD  - 

BCDA  - 

shows  that  the  tangents  at  B  and  D  meet  the  axis  of  the  projec- 
tivity where  it  is  met  by  the  line  A  C.  It  follows  that  the  diagonals 
of  the  quadrangle  ABCD  are  conjugate  lines  with  respect  to  the 
conic  (Fig.  77).  Hence  the  points  A,  B,  C,  D  must  be  harmonic 
points  (Art.  88)  in  order  to  form  one  cycle  of  a  projectivity. 

Conversely,  if  ABCD  are  any  four  harmonic  points  of  a  conic, 
it  is  easy  to  show  that  the  projectivity  determined  by  the  corre- 
spondence 

•     ABC  - 
BCD  - 

is  a  cyclic  projectivity  of  order  4  (cf.  Art.  22). 

4.  Cyclic  Projectivities  of  Higher  Order.  —  The  following  theorem 
holds  for  cyclic  projectivities  of  higher  order: 

Every  cyclic  projectivity  of  order  greater  than  2  is  necessarily 
elliptic. 

Let 

777        ITT  ET  77T  TJT 

111,   El,   -D2,   -C<3,  --  ,   Jiin-\ 

be  one  cycle  of  a  cyclic  projectivity  of  order  n>2.  No  two  con- 
secutive elements  of  this  cycle  can  be  separated  by  any  other  pair 
of  elements  of  the  cycle.  For,  suppose  E  and  E\  are  separated  by 
E2  and  Es,  so  that 


is  an  arrangement  of  the  elements  of  the  cycle  such  that  no  two 
consecutive  elements  are  separated  by  any  other  pair.  But  this 
arrangement  is  projectively  related  to  the  arrangement 

El,  ES,   E-2,  En,  -  ,   En-l,  E, 

in  which  E  and  EI  are  not  separated  by  E2  and  ^3.  This  contra- 
dicts the  hypothesis,  and  we  conclude,  therefore,  that  no  two  con- 
secutive elements  of  a  cycle  can  ever  be  separated  by  any  other 
pair  of  elements  of  the  cycle  (cf.  Art.  39). 


126  PROJECTIVE  GEOMETRY  [§101 

As  a  variable  element  describes  continuously  the  cycle 
E,  EI,  EZ, ,  En-\ 

its  corresponding  element  describes  continuously  the  correspond- 
ing arrangement 

E\,  EZ,  E$,  Et, ,  En-i,  E', 

and  it  is  obvious  that  the  two  can  never  coincide.  Consequently 
there  are  no  double  elements,  and  the  projectivity  is  elliptic. 
Note  that  the  argument  does  not  apply  when  n  =  2. 

Exercises 

1.  Given  three  points  A,  B,  C,  of  a  conic,  construct  the  fourth 
harmonic  point. 

2.  Given  three  tangents  a,  b,  c,  to  a  conic,  construct  the  fourth 
harmonic  tangent. 

3.  A  sheaf  of  rays  of  the  first  order  is  protectively  related  to  a 
sheaf  of  rays  of  the  second  order.     Show  how  to  construct  a  chain  of 
perspectivity    connecting    the   two   forms.     If   the   two    forms    are 
coplanar,  they  generate  a  curve  of  higher  order;  construct  a  number  of 
points  of  this  curve. 

4.  A  projectivity  on  a  conic  is  determined  by  three  pairs  of  corre- 
sponding points.     Construct  a  number  of  pairs  of  corresponding  points. 
If  corresponding  points  are  joined  by  straight  lines,  what  can  be  said 
about  these  lines? 

6.  A  projectivity  on  a  point-row  of  first  order  is  determined  by 
three  pairs  of  corresponding  points.  Construct  the  double  points, 
if  these  exist. 

6.  Construct  a  parabolic  projectivity  on  a  conic;  on  a  straight  line. 

7.  Construct  the  projectivity  on  a  conic  of  which  is  given  one  pair 
of  corresponding  points  and  the  axis,  the  axis  being  the  ideal  line  in 
the  plane. 

8.  Construct  a  cyclic  projectivity  of  order  3  on  a  conic,  the  axis 
being  the  ideal  line  in  the  plane.     Is  the  construction  possible  if  the 
conic  is  a  parabola  or  a  hyperbola?     Why? 


THE  THEORY  OF  INVOLUTION— IMAGINARY  ELEMENTS 

102.  Definition  of  Involution. — The  cyclic  pro jectivi ties  of  order 
2  have  a  special  nomenclature  and  a  special  theory  principally  on 
account  of  their  importance  in  the  geometry  of  the  eight  forms. 

A  cyclic  projectivity  of  order  2  on  any  form  is  called  an  involu- 
tion on  that  form.  Thus,  two  protectively  related  and  superposed 
forms  constitute  an  involution  on  their  common  base  if  the  pro- 
jectivity they  determine  is  cyclic  and  of  order  2. 

When  an  involution  exists  on  any  form,  the  elements  of  the 
form  are  said  to  be  paired  in  involution  or  to  be  doubly  corre- 
sponding. The  correspondence  between  the  elements  is  called 
an  involutoric  correspondence,  and  the  form  upon  which  the 
involution  exists  is  said  to  be  in  involution. 

Since  an  involution  is  a  projectivity,  it  may  have  two,  one,  or 
no  double  elements.  The  double  elements  of  an  involution  are 
called  focal  elements.  Thus,  the  double  points  of  an  involution 
on  a  point-row  of  first,  or  of  second,  order  are  the  foci  of  the  involu- 
tion; the  double  rays  of  an  involution  on  a  sheaf  of  rays  of  first,  or 
of  second,  class  are  the  focal  rays  of  the  involution;  and  the  double 
planes  of  an  involution  on  a  sheaf  of  planes  of  first,  or  of  second, 
class  are  the  focal  planes  of  the  involution. 

An  involution  on  any  form  is  hyperbolic,  parabolic,  or  elliptic, 
according  as  it  has  two,  one,  or  no  focal  elements. 

103.  Fundamental  Theorems. — The  theorems  in  Art.  101,  1, 
are  fundamental  for  the  theory  of  involution.     They  may  be 
restated  as  follows: 

1.  An  involution  on  any  form  is  completely  determined  by  two 
pairs  of  corresponding  elements. 

2.  Theorem   XIII. — The    straight    lines   joining    corresponding 
points  in  an  involution  on  a  conic  intersect  in  one  and  the  same  point 
U,  called  the  center  of  the  involution;  and  tangents  at  corresponding 

127 


128  PROJECTIVE  GEOMETRY  [§104 

points  meet  upon  one  and  the  same  straight  line  u,  called  the  axis 
of  the  involution.  The  center  and  the  axis  are  pole  and  polar  line 
with  respect  to  the  conic.  The  axis  meets  the  conic  in  the  foci  of  the 
involution.  The  involution  is  hyperbolic,  parabolic,  or  elliptic 
according  as  the  center  is  outside,  on,  or  inside  the  conic. 

Since  an  involution  on  any  form  can  be  reduced  by  projection 
or  section  to  an  involution  on  a  conic  (Art.  97),  the  proofs  in 
the  following  articles  will  be  confined  to  involutions  on  a  conic. 

104.  Hyperbolic  Involutions. — 

1.  //  an  involution  is  hyperbolic,  no  pair  of  corresponding  ele- 
ments is  separated  by  any  other  pair  of  corresponding  elements;  and 
conversely. 

For  the  lines  joining  corresponding  points  in  an  hyperbolic 

involution  on  a  conic  meet  in  the  ex- 
ternal center.  Hence,  no  two  pairs 
of  points  in  the  involution  can  sepa- 
rate each  other.  Conversely,  if  two 
pairs  of  points  in  a  given  involution 
on  a  conic  do  not  separate  each  other, 
the  center  of  the  involution  is  outside 
the  conic,  and  the  involution  is 
hyperbolic. 

2.  Any  pair  of  corresponding  ele- 
ments in  a  hyperbolic  involution  is 
separated  harmonically  by  the  focal 
elements  of  the  involution. 

For,  if  A  and  AI  are  a  pair  of  corresponding  points  in  an  hyper- 
bolic involution  on  a  conic  (Fig.  78),  the  line  AAi  is  conjugate 
to  the  axis  MN  with  respect  to  the  conic.  The  points  M,  A, 
N,  AI  are,  therefore,  harmonic  (Art.  88).  Similarly,  M,  B,  N, 
BI  are  four  harmonic  points. 

105.  Elliptic  Involutions. — 

1.  //  an  involution  is  elliptic,  any  pair  of  corresponding  elements 
is  separated  by  every  other  pair  of  corresponding  elements;  and 
conversely. 

For  the  lines  joining  corresponding  points  of  an  elliptic  in- 
volution on  a  conic  meet  in  the  internal  center.  Consequently 
every  pair  of  points  in  the  involution  is  separated  by  every  other 


§106]  ELEMENTARY  FORMS  129 

pair.  Conversely,  if  two  pairs  of  points  in  a  given  involution 
on  a  conic  separate  each  other,  the  center  of  the  involution  is 
inside  the  conic,  and  the  involution  is  elliptic. 

2.  Given  any  'pair  of  elements  in  an  elliptic  involution,  there 
exists  one,  and  but  one,  pair  of  elements  in  the  involution  which 
separates  harmonically  the  given  pair. 

Suppose  A,  A i  is  any  pair  of  points  in  an  elliptic  involution 
on  a  conic  (Fig.  79).  The  line  joining  the  pole  of  AAi  to  the 
center  of  the  involution  cuts  the  conic  in  the  pair  B,  BI.  A,  B, 
A  i,  BI  are  four  harmonic  points  (Art. 
88).  It  is  evident  that  the  line  PU 
must  cut  the  conic,  and  that  there 
is  but  one  such  line. 

106.  Parabolic  Involutions. — Since 
the  center  U  of  a  parabolic  involution 
on  a  conic  is  itself  a  point  of  the  conic, 
it  follows  that  U  forms  one  point  of 
each  pair  in  the  involution.  It  will 
be  seen,  then,  that  a  parabolic  involu- 
tion is  not,  in  the  true  sense  of  the  definition,  a  projectivity 
between  two  superposed  forms.  For  this  reason,  parabolic  in- 
volutions are  often  called  trivial  or  degenerate. 

Exercises 

1.  With  the  aid  of  Art.  104,  2,  show  that  the  foci  of  a  hyperbolic 
involution   on  a  straight  line  separate  harmonically    any  pair  of 
points  in  the  involution. 

2.  Prove  the  proposition  in  the  preceding  exercise  without  the  aid 
of  Art.  104,  2. 

Suggestions. — In  Fig.  80,  let  M  and  N  be  the  foci  and  A,  AI,  any 
pair  of  corresponding  points.  Then : 

(a)  ANAiM7iAiNAM;_Whyf 

(b)  S(ANA1M)^ROPM^A1NAM;  Why? 

(c)  RAi,  ON,  PA  meet  in  a  point  Q; 

(d)  ANA\M  is  a  harmonic  range.     Why? 

3.  An  involution  on  an  envelope  of  second  class  is  cut  by  any  ray 
of  the  envelope  in  an  involution  on  that  ray.     Conversely,  if  an  involu- 
tion is  chosen  arbitrarily  upon  any  ray  of  an  envelope,  the  rays  of  the 


130  PROJECTIVE  GEOMETRY  [§107 

envelope  passing  through  pairs  of  points  in  the  given  involution  are 
themselves  paired  in  involution.  Hence,  if  an  involution  is  chosen 
upon  a  tangent  to  a  conic,  show  that  tangents  drawn  from  pairs  of 
corresponding  points  intersect  upon  a  fixed  straight  line.  • 

4.  The  rays  of  a  sheaf  of  rays  of  first  class  are  paired  so  that  the 
rays  of  any  pair  are  mutually  perpendicular.     Show  that  the  sheaf  is 
in  involution. 

5.  The  vertices  of  all  right  angles  whose  sides  are  tangent  to  a  given 
parabola  lie  upon  a  fixed  straight  line,  and  the  lines  joining  the  points 
of  contact  on  a  pair  of  sides  pass  through  a  fixed  point. 

6.  The  axis  of  a  given  involution  on  a  conic  is  a  diameter  of  the 
conic.     Show  that  the  tangent  at  either  extremity  of  the  diameter  is 
cut  by  tangents  at  corresponding  points  of  the  involution  in  an  involu- 
tion having  one  focus  at  infinity.     Where  is  the  other  focus? 


FIG.  80. 

107.  Involutions  on  a  Straight  Line. — 

1.  If  0  is  the  mid-point  between  the  foci  M  and  N  of  a  hyperbolic 
involution  on  a  straight  line,  and  A,  A\  is  any  pair  of  points  in 
the  involution;  then 

OA-OAi  =  OM~2  =  ON*. 

For  MANAi  is  a  harmonic  range  of  points.  Hence  Art. 
30  applies.  The  point  0  corresponds  to  the  ideal  point  in  the 
involution. 

Again  (Fig.  81),  the  circles  on  AAi,  BB\,  CCi,  etc.,  as  diameters 
cut  the  circle  on  MN  orthogonally  (Art.  32). 

2.  An  elliptic  involution  on  a  straight  line  can  be  projected  from 
two  points  of  any  plane  through  the  line  in  sheaves  of  rays  in  in- 
volution such  that  the  rays  of  any  pair  are  mutually  at  right  angles. 

Draw  circles  on  AAi  and  BBi  as  diameters  (Fig.  82).     These 


§107] 


THE  THEORY  OF  INVOLUTION 


131 


circles  must  intersect  in  two  points  P  and  Q,  since  the  involution 
is  elliptic.  Project  the  involution  from  P  and  thus  obtain  a  sheaf 
of  rays  in  involution  in  which  the  corresponding  rays  PA  and  PA\t 
and  also  the  rays  PB  and  PBi,  are  perpendicular  to  each  other. 


FIG.  81. 

This  sheaf  is  cut  by  either  of  the  two  circles  in  an  involution  whose 
center  coincides  with  the  center  of  the  circle.  For  example,  the 
circle  PBBi  cuts  the  sheaf  in  an  involution  whose  center  is  R.  It 
follows  that  the  rays  of  any  pair,  in  the  involution  on  the  sheaf 


FIG.  82. 

whose  center  is  P,  are  perpendicular  to  each  other.  The  same 
statements  apply  to  the  point  Q.  The  points  P  and  Q  are,  then, 
the  two  points  from  which  the  involution  on  the  line  can  be  pro- 
jected as  stated  in  the  theorem. 

As  a  direct  consequence  of  this  theorem,  we  see  that  the  circles 


132  PROJECTIVE  GEOMETRY  [§108 

constructed  on  A  A  i,  BBi,  CC\,  etc.,  as  diameters,  all  pass  through 
the  points  P  and  Q. 

Again,  if  PQ  meets  AAi  in  0,  then  OP  is  a  mean  proportional 
between  the  segments  in  which  0  divides  the  diameter  of  any  one 
of  the  circles.  Since  these  segments  are  drawn  in  opposite  direc- 
tions from  0,  we  have 

OA-OAl  =  -OP2, 

where  A,  A\  is  any  pair  of  points  in  the  given  involution.  The 
point  0  corresponds  to  the  ideal  point  in  the  given  involution. 

108.  Involutions  on  a  Sheaf  of  Rays  of  First  Class. — //  the  rays 
of  a  sheaf  of  rays  of  first  class  are  paired  in  involution,  then  there  is 
at  least  one  pair  of  corresponding  rays  such- that  each  ray  is  perpen- 
dicular to  the  other.  If  there  is  more  than  one  such  pair,  then  the 
rays  of  every  pair  are  perpendicular  to  each  other. 

Consider  the  involution  determined  upon  a  circle  in  perspective 
position  with  the  given  sheaf.  That  diameter  of  the  circle  which 
passes  through  the  center  of  this  involution  cuts  the  circle  in  two 
points  that  are  projected  from  the  center  of  the  sheaf  in  a  pair  of 
corresponding  rays  such  that  each  ray  is  perpendicular  to  the 
other.  Obviously  there  is  one,  and  only  one,  such  pair  unless  the 
center  of  the  involution  on  the  circle  coincides  with  the  center  of 
the  circle.  But  this  will  be  the  case  if  there  are  two  pairs  of  mu- 
tually perpendicular  rays,  for  then  each  pair  will  cut  the  circle  at 
opposite  ends  of  a  diameter. 

An  involution  on  a  sheaf  of  rays  of  first  class  such  that  the  rays 
of  every  pair  are  at  right  angles  to  each  other  is  called  a  circular 
involution.  It  follows  from  Art.  107,  2,  that  an  elliptic  involu- 
tion on  any  form  can  be  reduced  by  projection  and  section  to  a 
circular  involution. 

Exercises 

1.  In  a  given  involution  on  a  straight  line,  0  corresponds  to  the 
ideal  point  and  the  corresponding  points  A  and  A\  are  situated  re- 
spectively two  units  and  eight  units  to  the  right  of  0.     Construct  a 
number  of  pairs  of  corresponding  points,  in  particular,  the  foci. 

2.  If,  in  the  preceding  exercise,  A  and  A\  are  situated  respectively 
two  units  to  the  right  and  eight  units  to  the  left  of  0,  construct  a 


U09] 


THE  THEORY  OF  INVOLUTION 


133 


number  of  pairs  of  corresponding  points.     Locate  the  points  from 
-which  the  given  involution  can  be  projected  in  a  circular  involution. 

3.  Given  the  foci  of  a  hyperbolic  involution  on  a  straight   line, 
construct  a  number  of  pairs  of  corresponding  points. 

4.  Given  the  foci  of  a  hyperbolic  involution  on  a  conic,  construct 
a  number  of  pairs  of  corresponding  points. 

6.  Given  an  involution  on  a  sheaf  of  rays  of  first  class,  construct 
the  pair  whose  rays  are  mutually  perpendicular. 

6.  Two  projectively  related  sheaves  of  rays  being  given,  how  can 
they  be  brought  into  such  a  position  as  to  form  an  involution? 

7.  If  a  right-angled  triangle  inscribed  in  a  conic  varies  so  that  its 
vertex  remains  fixed,  its  hypothenuse  will  constantly  pass  through  a 
fixed  point. 

8.  Two  fixed  points  A  and  B  are  chosen  on  a  fixed  tangent  to  a 
conic,  show  that  tangents  to  the  conic  from  points  harmonically 
separating  A  and  B  intersect  upon  a  fixed  straight  line. 

109.  Involutions  Determined  by  a  Complete  Quadrangle,  or  a 
Complete  Quadrilateral. — 


Opposite  sides  of  a  complete 
quadrangle  are  cut  by  a  line  in 
pairs  of  points  of  an  involution. 


Opposite  vertices  of  a  complete 
quadrilateral  are  projected  from 
a  point  in  pairs  of  rays  of  an 
involution. 


FIG.  83. 

For  the  theorem  on  the  left,  let  u  (Fig.  83)  cut  the  sides  RT  and 
SQ,  ST  and  QR,  QT  and  RS,  in  the  points  A  and  Aly  B  and  BI, 
C  and  Ci,  respectively,  and  let  0  be  the  point  of  intersection  of  QS 


134  PROJECTIVE  GEOMETRY  [§110 

and  RT.     Project  the  range  ATOR,  first  from  Q  and  then  from  S 
and  cut  the  resulting  pencils  of  rays  by  the  line  u.     We  thus  obtain, 

ACAiBi  A  ATOR  A  ABA^d. 
But  ABAtCi  7\  A,C,AB  (Art.  101,  1).     Hence, 


and  consequently  the  involution  determined  by  the  doubly  cor- 
responding pair  A,  A  i  and  the  pair  B,  B\  will  contain  the  pair 
C,  C\.  This  proves  the  theorem. 

Exercises 

1.  Prove  the  theorem  on  the  right  by  means  of  the  principle  of 
duality.     Construct  the  figure. 

2.  In  the  theorem  on  the  left,  show  that,  if  u  cuts  the  quadrangle 
so  that  two  or  four  of  the  vertices  are  on  the  same  side  of  u  the  in- 
volution determined  upon  u  is  hyperbolic;  if  u  cuts  the  quadrangle  so 
that  one  vertex  is  on  one  side  and  three  vertices  on  the  other,  the 
involution  is  elliptic  ;  and  if  u  passes  through  a  vertex  of  the  quadrangle 
the  involution  is  parabolic. 

3.  In  the  theorem  on  the  right,  the  sides  of  the  quadrilateral  divide 
the  plane  into  regions.     In  which  regions  must  a  point  be  placed  so 
that  the  involution  determined  by  the  quadrilateral  shall  be  hyper- 
bolic?    Elliptic?     Parabolic? 

4.  Given  two  pairs  of  points,  A,  AI  and  B,  Bi,  in  an  involution  on 
a  straight  line.     Show  how  to  construct  the  point  corresponding  to 
C  in  the  involution  by  means  of  a  complete  quadrangle. 

110.  Desargues  Theorem  and  Its  Dual.  — 

The  points  of  intersection  of  a          The  tangents  drawn  from  a 
fixed   straight   line  with   conies     fixed  point  to  conies  inscribed  in 


circumscribed  about  a  complete 
quadrangle  form  pairs  of  points 


a   complete    quadrilateral   form 
pairs  of  rays  in  the  involution 


in  the  involution  determined  upon  j  determined  about  the   point   by 
the  line  by  the  quadrangle.  \  the  quadrilateral. 

For  the  theorem  on  the  left,  let  QRST  be  the  complete  quad- 
rangle (Fig.  84),  and  u  the  fixed  straight  line -meeting  the  sides 
TQ,  QR,  RS,  and  ST  in  the  points  A,  B,  AI,  and  B\,  respectively. 
Let  any  conic  circumscribed  about  the  quadrangle  meet  u  in  the 
points  P  and  Pj.  The  pencils  S(TPRPJ  and  Q(TPRP^  are 
projectively  related  since  corresponding  rays  meet  on  the  conic. 
Hence, 


§111] 


THE  THEORY  OF  INVOLUTION 


135 


i  X  APBPl  X  BPiAP  (Art.  101,  1);  ' 

and  therefore  P  and  P\  are  doubly  corresponding  points  in  the  in- 
volution in  which  A,  A\  and  B,  BI  are  two  pairs  of  corresponding 
points.  But  this  involution  is  determined  by  the  complete  quad- 
rangle QRST. 

Exercises 

1.  Draw  a  figure  and  prove  the  theorem  on  the  right. 

2.  A  hyperbolic  involution  on  a  straight  line  is  determined  by  a 
complete  quadrangle,  construct  the  foci  of  the  involution. 

3.  A    hyperbolic    involution 
about  a  point  is  determined  by 
a  complete  quadrilateral,  con- 
struct  the  focal    rays    of    the 
involution. 

4.  In  general,  two  conies  can 
be  drawn    through  four  given 
points  and  tangent  to  a  given 
line.     Construct  the  points  of 
contact  with  the  given  line. 

6.  How  many  parabolas  can 
be   drawn  through  four  given 

points  in  a  plane?  Under  what  circumstances  are  there  no  parabolas 
through  four  given  points?  • 

6.  Construct  the  axes  of  the  parabolas  that  can  be  drawn  through 
four  given  points. 

7.  In  general,  two  conies  can  be  drawn  to  touch  four  given  lines 
and  pass  through  a  given  point.     Construct  the  tangents  to  the  conies 
at  the  given  point. 

8.  How  many  parabolas  can  be  drawn  to  touch  the  sides  of  a  given 
triangle  and  pass  through  a  given  point  ?    Construct  the  points  of  con- 
tact of  the  parobolas  that  touch  the  sides  of  the  triangle  ABC  and 
pass  through  the  point  P. 

111.  Involutions  Determined  by  a  Fixed  Conic. — 


FIG.  84. 


A  fixed  conic  determines  upon 
any  straight  line  in  its  plane  an 
involution  such  that  the  points 
of  each  pair  are  conjugate  with 
respect  to  the  conic. 


A  fixed  conic  determines  about 
any  point  in  its  plane  an  invo- 
lution, such  that  the  rays  of  each 
pair  are  conjugate  with  respect 
to  the  conic. 


136  PROJECTIVE  GEOMETRY  [§111 

Any  straight  line  in  the  plane  of  a  fixed  conic  is  the  support  of 
two  point-rows  so  related  that,  with  respect  to  the  conic,  any  point 
of  either  is  conjugate  to  some  point  of  the  other.  Either  point-row 
is  a  section  of  the  sheaf  of  polar  lines  corresponding  to  the  other. 
The  two  point-rows  are  therefore  projectively  related  (Art.  69,  1) 
and  thus  constitute  a  projectivity  upon  their  common  base.  Since 
corresponding  points  are  conjugate  with  respect  to  the  conic,  the 
projectivity  is  cyclic  and  of  order  2.  It  is,  therefore,  an  involution. 

Similarly,  any  point  in  the  plane  of  a  fixed  conic  is  the  support 
of  two  projectively  related  sheaves  of  rays,  corresponding  rays 
being  conjugate  with  respect  to  the  conic.  The  two  sheaves  thus 
constitute  an  involution  about  their  common  center. 

The  foci  of  the  involution  determined  upon  any  line  by  a  fixed  conic 
are  the  points  in  which  the  conic  meets  the  line. 

For  the  points  common  to  the  line  and  the  conic  are  self- 
conjugate  with  respect  to  the  conic. 

The  involution  is,  therefore,  hyperbolic,  parabolic,  or  elliptic 
according  as  the  line  meets  the  conic  in  two  distinct  points,  is 
tangent  to  the  conic,  or  does  not  meet  the  conic  at  all. 

The  focal  rays  of  the  involution  determined  about  any  point  by  a 
fixed  conic  are  the  tangents  which  can  be  drawn  from  the  point  to  the 
conic. 

For  these  tangents  are  self-conjugate  with  respect  to  the  conic. 
The  involution  is  therefore  hyperbolic,  parabolic,  or  elliptic  ac- 
cording as  the  point  is  outside,  on,  or  inside,  the  conic.  In  par- 
ticular, if  the  point  is  the  center  of  the  conic,  the  involution  con- 
sists of  pairs  of  conjugate  diameters.  If  the  conic  is  a  hyperbola, 
the  focal  rays  are  the  asymptotes.  Since  there  is  always  at  least 
one  pair  of  rays  at  right  angles  in  every  involution  on  a  sheaf  of  rays 
(Art.  108),  we  see  that  there  must  always  be  at  least  one  pair  of 
conjugate  diameters  at  right  angles  (cf.  Art.  76).  If  there  is  more 
than  one  such  pair,  the  conic  is  necessarily  a  circle. 

Exercises 

1.  If  an  involution  on  a  conic  is  projected  from  any  point  of  the 
conic  upon  the  axis  of  the  involution,  show  that  the  resulting  involu- 
tion on  the  axis  consists  of  pairs  of  points  conjugate  with  respect  to 
the  conic  (cf.  Art.  69,  3). 


§112]  THE  THEORY  OF  INVOLUTION  137 

2.  Write  out  and  prove  the  dual  of  exercise  1. 

3.  Given  two  pairs  of  points;  A,  A\  and  B,  B\;  of  an  involution  on 
a  straight  line.     By  means  of  a  complete  quadrangle,  construct  the 
point  Ci  corresponding  to  any  fifth  point  C;  and  in  particular,  the 
point  corresponding  to  the  ideal  point  of  the  line. 

4.  Two  pairs  of  conjugate  diameters  of  a  conic  are  known,  draw 
the  conjugate  to  any  fifth  diameter  and  construct  the  axes. 

6.  If  two  pairs  of  opposite  sides  of  a  complete  quadrangle  are  at 
right  angles,  show  that  the  third  pair  is  also  at  right  angles. 

6.  By  the  preceding  exercise,  show  that  the  perpendiculars  let 
fall  from  the  vertices  of  a  triangle  upon  the  sides  opposite  meet  in  a 
point.     This  point  is  called  the  orthocenter  of  the  triangle. 

7.  Prove  that  all  conies  which  pass  through  the  vertices  and  the 
orthocenter  of  a  triangle  are  equilateral  hyperbolas. 

8.  Show  that  an  equilateral  hyperbola  can  always  be  circumscribed 
about  any  quadrangle.     This  hyperbola  passes  through  the  ortho- 
centers  of  the  four  triangles  formed  by  the  vertices  of  the  quadrangle. 

9.  The  sides  of  any  triangle  form  with  the  ideal  line  in  the  plane  a 
complete  quadrilateral  whose  three  pairs  of  opposite  vertices  are  pro- 
jected from  the  orthocenter  of  the  triangle  in  three  pairs 'of  rays  at 
right  angles. 

10.  By  the  preceding  exercise,  show  that  the  two  tangents  which 
can  be  drawn  from  the  orthocenter  of  any  triangle  circumscribed 
about  a  parabola  are  at  right  angles  to  each  other;  and  hence  the 
orthocenters  of  all  triangles  circumscribed  about  a  parabola  lie  on  a 
fixed  straight  line. 

11.  How  does  the  preceding  exercise  show  that  the  orthocenters  of 
the  four  triangles  formed  by  the  sides  of  any  complete  quadrilateral 
lie  on  one  straight  line? 

12.  If  a  circle  is  circumscribed  about  the  rectangle  ABCD,  then  the 
tangents  drawn  from  any  point  S  of  this  circle  to  any  conic  inscribed 
in  the  rectangle  are  at  right  angles  to  each  other. 

13.  The  vertices  of  all  right  angles  whose  sides  touch  an  ellipse  or 
a  hyperbola  lie  upon  a  circle. 

14.  If  any  two  of  the  three  circles  which  have  the  diagonals  of  a 
complete  quadrilateral  for  diameters  intersect;  then  the  third  circle 
passes  through  the  points  of  intersection. 

15.  If  a  quadrangle  is  inscribed  in  a  circle,  show  that  all  hyperbolas 
passing  through  the  vertices  of  the  quadrangle  have  parallel  axes. 

112.  Imaginary  Points  on  a  Straight  Line.— A  hyperbolic  in- 
volution on  a  straight  line  defines  two  real  points;  namely,  the  foci 


138  PROJECTIVE  GEOMETRY  [§113 

of  the  involution.  For  the  foci  can  be  constructed  as  soon  as  we 
know  two  pairs  of  points  of  the  involution. 

Since  the  involution  is  determined  by  any  two  pairs  of  corre- 
sponding points,  say  A,  A\  and  B,  BI,  we  may  distinguish  between 
the  foci  by  the  following  convention :  reading  the  points  in  an  order 
such  that  the  pair  B,  BI  lies  to  the  right  of  the  pair  A,  AI  shall 
define  that  focus  which  separates  B,  BI  from  either  A  or  AI;  and 
reading  the  points  in  reverse  order,  so  that  the  pair  A,  A i  lies  to  the 
right  of  B,  BI,  shall  define  the  focus  that  separates  the  pair  A,  AI 
from  either  B  or  B  i.  Hence  we  can  say : 

Two  pairs  of  points  A,  A  i  and  B,  BI,  lying  on  the  same  straight  line, 
but  not  separating  each  other,  together  with  the  direction  in  which  they 
are  read,  serve  to  define  absolutely  a  real  point  of  the  line  which  is  one 
focus  of  the  hyperbolic  involution  determined  by  the  given  pairs. 

In  an  entirely  analogous  manner,  we  shall  say  that  an  elliptic  in- 
volution on  a  straight  line  defines  two  conjugate  imaginary  points 
which  may  be  thought  of  as  the  foci  of  the  involution. 

An  elliptic  involution  is  determined  by  two  pairs  of  points  A, 
AI  and  B,  BI,  which  separate  each  other;  and  we  shall  distinguish 
between  the  imaginary  points  which  the  involution  defines  by 
the  direction  in  which  the  given  pairs  are  read.  Thus,  ABA\B\ 
defines  one  imaginary  point  while  B\A\BA  defines  the  conjugate 
imaginary  point.  Hence: 

Two  pairs  of  points  A,  AI  and  B,  B\,  lying  on  the  same  straight 
lines,  and  separating  each  other,  together  with  the  direction  in  which 
they  are  read,  define  an  imaginary  point.  The  conjugate  imaginary 
point  is  defined  by  reading  the  given  pairs  in  reverse  order. 

As  an  immediate  consequence  of  this  definition: 

Two  conjugate  imaginary  points  are  joined  by  the  real  line  on 
which  lies  the  elliptic  involution  defining  them. 

113.  Imaginary  Lines  in  a  Plane. — An  elliptic  involution  on  a 
sheaf  of  rays  of  first  class  defines  two  conjugate  imaginary  lines  in 
the  plane  of  the  sheaf.  Since  an  elliptic  involution  on  a  sheaf  of 
rays  of  first  class  is  determined  by  two  pairs  of  rays  which  separate 
each  other  we  have  the  definition : 

Two  pairs  of  rays,  a,  a\  and  b,  bi  of  a  sheaf  of  rays  of  first  class 
which  separate  each  other,  together  with  the  order  in  which  they  are 


§115] 


THE  THEORY  OF  INVOLUTION 


139 


taken,  define  an  imaginary  line.  The  conjugate  imaginary  line 
is  defined  by  taking  the  given  pairs  in  reverse  order. 

As  a  consequence  of  this  definition: 

Two  conjugate  imaginary  lines  in  the  same  plane  always  intersect 
in  a  real  point.  This  point  is  the  support  of  the  sheaf  of  rays  of  first 
class  upon  which  lies  the  elliptic  involution  defining  the  imaginary 
lines. 

114.  Imaginary  Planes. — An  elliptic  involution  on  a  sheaf  of 
planes  of  first  class   defines  two   conjugate  imaginary  planes. 
These  imaginary  planes  intersect  in  the  axis  of  the  sheaf  of  planes. 
If  the  involution  is  determined  by  the  two  pairs  of  planes  a,  ot\  and 
j3,  |3i,  then  we  shall  say  that  the  order  aftctifti  defines  one  of  the 
imaginary  planes  while  the  reverse  order  fiictifia  defines  the  con- 
jugate imaginary  plane. 

115.  Construction  Problems. 


To  construct  the  point  of 
intersection  of  a  real  line  with  an 
imaginary  line. 

fi  be  the  imaginary  point 
S 


1 .  To  construct  the  line  joining 
a   real   point   to   an   imaginary 
point. 

Let  S  be  the  real  point  and  let 
(Fig.  85).  Project  ABArf^ 
from  S  and  thus  obtain  the 
imaginary  line  aba\bi.  This 
is  the  line  to  be  constructed. 
The  conjugate  imaginary  line 
biaiba  joins  the  conjugate  im- 
aginary point  BiAiBA  to  S. 

If  u  is  a  real  line  and  aba\bi 
an  imaginary  line  lying  in  the 
same  plane,  they  intersect  in 
the  imaginary  point  ABAiBi. 
The  conjugate  imaginary  line 
intersects  u  in  the  conjugate  imaginary  point  BiAiBA. 

If  we  project  an  imaginary  line  from  a  real  point  not  lying  in  the 
plane  of  its  defining  sheaf,  we  obtain  an  imaginary  plane;  and  if  we 
cut  an  imaginary  plane  by  a  real  plane  not  passing  through  the  axis 
of  its  defining  sheaf,  we  obtain  an  imaginary  line. 

2.  To  construct  the  line  joining  two  imaginary  points  in  a  plane. 


FIG.  85. 


140  PROTECTIVE  GEOMETRY  [§115 

Let  two  imaginary  points  be  defined  by  elliptic  involutions  along 
the  lines  u  and  u'  (Fig.  86).  Let  A  be  the  point  common  to  u  and 
u'  and  let  A\  correspond  to  A  in  the  involution  on  u,  and  A\ 
correspond  to  A  in  the  involution  on  u'.  Construct  the  pair  B,  BI 
belonging  to  the  involution  and  separating  the  pair  A,  A\  harmon- 
ically (Art.  105,  2).  Similarly,  construct  the  pair  B',  B\  separating 


FIG.  86. 


A,  A' i  harmonically.  The  harmonic  ranges  ABAiB\  and  AB'- 
A'lB'i  are  sections  of  one  and  the  same  pencil  a'b'a'ib'i.  The 
elliptic  involution  determined  by  the  pairs  of  rays  a',  a'i  and 
ft1,  ft1!,  defines  the  imaginary  line  joining  the  given  imaginary 
points.  The  conjugate  imaginary  points  are  joined  by  the  conju- 
gate imaginary  line. 

The  imaginary  point  on  one  line  can  be  joined  to  the  conjugate 
imaginary  point  on  the  other.  Thus,  the  harmonic  ranges 
ABAiBi  and  AB\A\B'  are  sections  of  the  pencil  aba\bi  whose 
center  is  Q.  The  pairs  a,  Ci  and  6,  &i  define  the  line  sought. 

Similarly,  if  two  imaginary  lines  are  defined  by  elliptic  involu- 
tions on  sheaves  of  rays  whose  centers  are  P  and  Q,  we  can  select 
harmonically  separated  pairs  of  rays  and  so  obtain  harmonic  pen- 


§117] 


THE  THEORY  OF  INVOLUTION 


141 


cils  in  perspective  position  as  in  the  figure.  When  the  sheaves  are 
taken  in  the  same  order,  they  define  lines  intersecting  in  the  imagi- 
nary point  ABAiBi  or  its  conjugate  imaginary  point.  When  the 
sheaves  are  taken  in  opposite  orders,  they  define  lines  intersecting 
in  the  imaginary  point  AB'A\B\  or  in  its  conjugate  imaginary 
point. 

116.  Imaginary  Elements  on  Any  Form. — An  elliptic  involution 
on  any  form  defines  a  pair  of  conjugate  imaginary  elements  on 
that  form.  Thus,  an  elliptic  involution  on  a  conic  defines  a  pair 
of  conjugate  imaginary  points  on  the  conic.  These  points  are 
joined  by  the  axis  of  the  involution  (Art.  Ill,  exercise  1). 

Reciprocally,  an  elliptic  involution  on  an  envelope  defines  a  pair 
of  conjugate  imaginary  lines  belonging  to  the  envelope.  These 
lines  intersect  in  the  center  of  the  involution. 

The  introduction  and  definition  of  imaginary  elements  enables 
us  to  say: 


A  conic  meets  every  real  line  in 
its  plane  in  two  points  which  are 
real  and  distinct,  coincident,  or 
conjugate  imaginary,  according 
as  the  involution  determined  up- 
on the  line  by  the  conic  is  hyper- 
bolic, parabolic,  or  elliptic. 


A  conic  has  two  tangents  from 
every  real  point  in  its  plane  which 
are  real  and  distinct,  coincident, 
or  conjugate  imaginary,  accord- 
ing as  the  involution  determined 
about  the  point  by  the  conic  is  hy- 
perbolic, parabolic,  or  elliptic. 


By  projection,  these  statements  are  at  once  carried  over  to  cor- 
responding statements  concerning  the  cone  and  the  sheaf  of  planes 
of  second  class. 

An  elliptic  involution  on  a  regulus  defines  a  pair  of  conjugate  im- 
aginary lines  which  do  not  lie  in  any  real  plane  nor  pass  through  any 
real  point. 

Conjugate  imaginary  lines  defined  by  an  elliptic  involution  on 
a  regulus  are  often  called  conjugate  imaginary  lines  of  the  second 
kind.  Conjugate  imaginary  lines  of  the  first  kind  always  lie  in 
a  real  plane  and  intersect  in  a  real  point. 

117.  Construction  Problems. — 

1.  To  construct  the  conic  passing  through  five  points  of  a  plane, 
three  of  the  given  points  being  real  and  two  being  conjugate  imaginary 
points. 


142 


PROJECTIVE  GEOMETRY 


[§117 


Let  P,  Q,  and  R  be  the  real  points,  and  let  the  conjugate  imagi- 
nary points  be  denned  by  an  elliptic  involution  on  the  line  u 
(Fig.  87).  Let  PQ  meet  u  in  A,  and  QR  meet  u  in  B.  To  A  cor- 
responds a  point  A  i  in  the  involution,  and  to  B  corresponds  a 
point  BI.  The  pairs  A,  A\  and  B,  B\  must  necessarily  separate 
each  other,  since  the  involution  is  elliptic.  These  pairs  are  also 
conjugate  pairs  of  points  with  respect  to  the  conic  which  we  are  to 
construct.  Consequently,  the  polar  lines  of  A  and  B  pass  through 
AI  and  BI  respectively.  But  the  polar  line  of  A  passes  through  S, 
the  harmonic  conjugate  of  A  with  respect  to  P  and  Q,  and  the 


FIG.  87. 

polar  line  of  B  passes  through  T,  the  harmonic  conjugate  of  B 
with  respect  to  R  and  Q.  Therefore  these  polar  lines  can  be  con- 
structed. They  intersect  in  U,  the  pole  of  u  with  respect  to  the 
conic.  Let  the  line  R  U  meet  u  in  L,  and  the  line  PU  meet  u  in  K. 
The  conic  must  pass  through  M  and  N,  the  harmonic  conjugates 
of  R  and  P  with  respect  to  the  pairs  U,  L  and  U,  K,  respectively. 
We  thus  have  five  real  points,  P,  Q,  R,  M,  N,  through  which  the 
conic  must  pass.  Other  points  of  the  conic  can  now  be  constructed 
by  previous  methods. 

2.  To  construct  the  conic  passing  through  one  real  point  and  two 
pairs  of  conjugate  imaginary  points,  no  three  of  the  given  points 
lying  on  the  same  straight  line. 


§117] 


THE  THEORY  OF  INVOLUTION 


143 


Let  P  be  the  real  point,  and  let  the  pairs  of  conjugate  imaginary 
points  be  defined  by  elliptic  involutions  on  the  lines  u  and  u' 
(Fig.  88).  We  may  suppose  that  these  involutions  are  determined 
by  pairs  of  points  which  separate  each  other  harmonically  (Art. 
105,  2).  Let  these  pairs  be  A,  A\  and  B,  BI  on  u,  and  A,  A'i 
and  B',  B\  on  u',  where  A  is  the  point  common  to  u  and  u'.  Since 
corresponding  points  in  the  involutions  on  u  and  u'  are  conjugate 
with  respect  to  the  conic  we  are  to  construct,  the  polar  line  of  A 
must  pass  through  both  A\  and  A\.  If  the  line  PA  meets  the 


FIG.  88. 


polar  line  of  A  in  R,  then  the  harmonic  conjugate  of  P  with  respect 
to  A  and  R  must  lie  on  the  curve.  Let  Q  be  this  point.  We  can 
now  construct  the  quadrangle  PQST  whose  pairs  of  opposite  sides 
intersect  in  A  and  A\  and  whose  diagonals  pass  through  B  and  E\. 
This  quadrangle  is  inscribed  in  the  conic.  Similarly,  we  can  con- 
struct the  quadrangle  PQS'T'  which  is  also  inscribed  in  the  conic. 
We  thus  have  six  real  points  on  the  conic,  and  as  many  more  can 
be  constructed  as  may  be  desired  by  previously  given  methods. 

3.  Given  two  involutions  lying  upon  the  same  form;  to  construct 
the  pair  of  elements  belonging  to  both  involutions. 

Suppose  the  two  involutions  are  upon  the  same  conic,  and  let 
U  and  V  be  the  centers  of  the  involutions  (Fig.  89).  The  line  UV 


144  PROJECTIVE  GEOMETRY 

cuts  from  the  conic  the  pair  of  points  required,  since  these  points 
obviously  correspond  to  each  other  in  both  involutions. 

The  pair  of  points  common  to  two  given  involutions  on  the  same 
conic  will  be  real  if  the  involutions  are  both  elliptic,  or  if  one  is 
elliptic  and  the  other  is  hyperbolic,  as  in  the  figure.  For,  in  these 
cases,  the  line  UV  must  cut  the  conic  in  real  points. 

If  both  the  given  involutions  are  hyperbolic,  their  common  pair 
of  points  will  be  real  or  conjugate  imaginary,  according  as  the  line 

joining  their  centers  cuts  the 
conic  in  real  points  or  in  con- 
jugate  imaginary  points. 

In  any  case,  the  pair  of  points 
common  to  two  given  involu- 
tions on  a  conic  is  defined  by 
the  involution  which  the  conic 
determines  upon  the  line  joining 
their  centers. 

4.  Given  an  involution  on  a 
straight  line;  to  construct  the 

pair  of  points   belonging  to  the  involution  which  separates   har- 
monically any  two  real  points  on  the  line. 

Let  M  and  N  be  the  two  real  points  on  the  line.  Project  the 
given  involution  and  the  points  M  and  N  from  a  point  S  not  on 
the  line.  We  thus  have  an  involution  about  S.  Consider  the 
rays  SM  and  SN  as  the  focal  rays  of  a  second  involution  about  S. 
When  we  cut  the  two  involutions  about  S  by  a  conic  passing 
through  S,  we  obtain  two  involutions  on  the  conic.  The  pair 
of  points  common  to  these  involutions  project  from  S  in  rays 
which  are  cut  by  the  given  line  in  the  pair  of  points  required. 

Exercises 

1.  Construct  the  conic  tangent  to  five  lines  in  the  plane  two  of 
which  are  conjugate  imaginary  lines. 

2.  Two  involutions  lie  on  the  same  straight  line,  construct  the  pair 
of  points  belonging  to  both  involutions. 

3.  Given  an  involution  on  a  sheaf  of  rays  of  first  class;  construct 
the  pair  of  rays  belonging  to  the  involution  and  separating  harmon- 
ically any  given  pair  of  rays. 


§117]  THE  THEORY  OF  INVOLUTION  145 

4.  Given  an  involution  on  a  straight  line;  construct  that  pair  of 
points  which  belongs  to  the  involution  and  is  bisected  by  a  given  point. 

5.  Through  a  given  point  draw  a  pair  of  lines  which  will  intercept 
equal  segments  on  two  given  lines.     Is  there  more  than  one  such  pair? 

6.  In  a  given  conic  inscribe  a  triangle  whose  sides  shall  pass  through 
three  given  points. 

Suggestion. — Consider  the  three  given  points  as  the  centers  of  three 
involutions  on  the  conic.  Let  A  correspond  to  Ai  in  the  first  involu- 
tion, A  i  to  A 2  in  the  second,  and  A 2  to  A 3  in  the  third.  As  A  describes 
the  conic,  A3  will  describe  a  protectively  related  conic.  Consider 
the  double  points  of  the  projectivity  so  determined.  Discuss  the 
problem  for  various  positions  of  the  given  points.  Can  the  problem 
be  generalized  for  any  polygon? 

7.  State  the  dual  of  the  preceding  exercise  and  show  how  the  con- 
struction can  be  made. 

8.  Show  that  every  circle  in  a  plane  passes  through  the  same  con- 
jugate imaginary  points  on  the  ideal  line. 

9.  Given  a  point-row  u  and  a  sheaf  of  rays  U  projectively  related 
to  each  other;  find  a  point  from  which  u  can  be  projected  in  a  sheaf 
equiangular  with  U. 

NOTE. — The  student  will  scarcely  fail  to  notice  the  difference  in 
character  between  the  problems  in  Chapter  XI  and  XII  and  the 
problems  in  previous  chapters.  The  earlier  problems,  with  two  ex- 
ceptions, are  problems  of  the  first  degree ;  that  is,  problems  admitting 
of  but  one  solution.  Such  problems  are  reducible  to  the  problem  of 
finding  an  element  jn  one  of  two  projectively  related  forms,  cor- 
responding to  a  given  element  in  the  other;  and  are  solved  by  means  of 
a  chain  of  perspectivity  connecting  the  two  forms.  The  necessary 
constructions  are  carried  out  with  the  aid  of  a  straight  edge,  or  ruler. 
The  exceptions  noted  above  are  the  problem  of  finding  a  pair  of  points 
harmonically  separating  two  given  pairs  (Art.  32,  exercise  7)  and  the 
problem  of  constructing  the  axes  of  an  ellipse  or  of  a  hyperbola  (Art. 
76).  Each  of  these  problems  has  two  solutions  and  each  requires 
the-use  of  a  circle.  The  solution  of  a  problem  of  the  first  degree  is 
comparable  to  the  solution  of  a  linear  equation. 

Problems  of  the  second  degree  admit  of  at  most  two  solutions. 
Such  problems  are  reducible  to  the  problem  of  constructing  the 
double  points  of  a  projectivity  on  a  conic,  or  the  foci  of  an  involution 
on  a  conic.  The  solution  of  a  problem  of  the  second  degree  is  equiva- 
lent to  the  solution  of  a  quadratic  equation. 

The  introduction  of  imaginary  elements  into  projective  geometry 
is  due  to  Von  Staudt  (1847). 

10 


CHAPTER  XIII 
THE  FOCI  AND  FOCAL  PROPERTIES  OF  CONICS 

118.  Definition   of   Focus. — We   have   seen    (Art.  Ill)    that 
a  conic  sets  up  about  any  point  in  its  plane  an  involution  such 
that  the  rays  of  any  pair  are  conjugate  lines  with  respect  to  the 
conic.     This  involution  has  at  least  one  pair  of  rays  at  right  angles 
(Art.   108).     These  rays  are  called  normal  conjugate  rays.     If 
the  involution  has  more  than  one  pair  of  normal  conjugate  rays, 
then  the  rays  of  every  pair  are  perpendicular  to  each  other;  and  the 
involution  is  circular  (Art.  108). 

//  an  involution  which  a  conic  determines  about  a  point  F  is  cir- 
cular, then  F  is  a  focus  of  the  conic. 

It  follows  from  this  definition  that  a  focus  is  necessarily  inside 
a  conic.  For  the  involution  set  up  about  a  point  outside  the  conic, 
or  on  the  conic,  is  certainly  not  circular. 

Again,  a  focus  must  be  on  an  axis  of  the  conic.  For,  if  F  is 
a  focus  and  C  is  the  center,  the  diameter  FC  is  not  perpendicular 
to  its  conjugate  chord  through  F,  unless  FC  is  an  axis. 

If  the  center  of  a  conic  is  a  focus,  then  every  diameter  is  per- 
pendicular to  its  conjugate  diameter,  and  the  conic  is  necessarily 
a  circle. 

Conversely,  the  center  of  a  circle  is  a  focus.  For  every  diameter 
of  a  circle  is  perpendicular  to  its  conjugate  diameter.  It  is 
evident  that  the  circle  can  have  but  this  one  focus,  since  the 
involution  determined  about  any  other  point  has  but  one  pair  of 
normal  conjugate  rays. 

119.  Construction  of  Foci. — 

1.  The  Ellipse. — Let  AB  and  CD  be  the  axes  of  an  ellipse 
(Fig.  90).  Draw  the  tangents  at  the  extremities  of  these  axes, 
thus  circumscribing  a  rectangle  KLMN  about  the  ellipse.  The 
segments,  AB  and  CD,  which  the  ellipse  determines  upon  its  axes, 
are  unequal  in  length.  For,  if  these  segments  were  equal  in  length, 

146 


§119] 


PROPERTIES  OF  CONICS 


147 


K 


then  KLMN  would  be  a  square  and  we  could  inscribe  a  circle 
in  it  which  would  have  four  points  and  the  tangents  at  them  in 
common  with  the  ellipse.  The  ellipse  would  then  coincide 
throughout  with  the  circle.  Hence,  the  segments  AB  and  CD 
are  unequal  in  length.  The  longer  of  the  two  segments  is  the 
major  axis  of  the  ellipse,  and  the  shorter  is  the  minor  axis,  or 
conjugate  axis. 

Suppose  AB,  the  major  axis,  is  parallel  to  the  sides  LM, 
KN  of  the  circumscribed  rectangle.  The  circle  drawn  with  LM 
as  a  diameter  cuts  AB  in  two 
real  points,  F  and  F\,  which  are 
foci  of  the  ellipse.  For  the  lines 
KL,  LM,  and  MN  are  the  sides  of 
a  triangle  circumscribed  about  the 
ellipse,  one  vertex  being  at  infinity. 
The  points  F  and  FI  are  conju- 
gate to  this  ideal  vertex  with  re- 
spect to  the  ellipse.  Consequently, 
the  lines  LF  and  MF  are  conju- 
gate with  respect  to  the  ellipse 
(Art.  69,  3),  and  thus  form  one  pair 
of  normal  conjugate  rays  in  the  in- 
volution set  up  about  F.  The 
axis  A B  and  its  conjugate  chord 
through  F  form  a  second  pair  of  normal  conjugate  rays  in  this 
involution.  Consequently,  the  involution  about  F  is  circular 
and  F  is  a  focus.  Similarly,  FI  is  a  focus. 

The  segment  of  any  tarfgent,  contained  between  the  tangents 
at  A  and  B,  subtends  a  right  angle  at  F  or  at  FI.  Thus,  P'FQ' 
is  a  right  angle,  since  P'F  and  Q'F  are  conjugate  lines  with  respect 
to  the  ellipse  and  consequently  form  a  pair  of  rays  in  the  involution 
set  up  about  F  by  the  ellipse.  Hence,  the  circle  on  P'Q'  as  a 
diameter  passes  through  F.  Similarly,  this  circle  also  passes 
through  FI.  In  this  way  is  obtained  a  system  of  circles  each 
of  which  passes  through  the  foci  and  cuts  the  tangents  at  A 
and  B  at  opposite  ends  of  diameters. 

It  is  clear,  from  the  foregoing  discussion,  that  F  and  F\  are 
the  only  foci  on  the  major  axis  of  the  ellipse.  Since  the  circle 


FIG.  90. 


148 


PROJECTIVE  GEOMETRY 


[§119 


on  KL  as  a  diameter  cannot  cut  the  minor  axis  in  real  points, 
it  follows  that  there  are  no  real  foci  on  the  minor  axis.  Conse- 
quently, F  and  FI  are  the  only  real  foci  of  the  ellipse. 

2.  The  Hyperbola. — Only  one  of  the  axes  of  a  hyperbola 
meets  the  curve  in  real  points.  This  is  the  major  axis  AB  (Fig. 
91).  Draw  tangents  to  the  curve  at  A  and  B  meeting  the 
asymptotes  in  the  points  K,  M  and  L,  N,  respectively.  The  circle 
circumscribed  about  the  rectangle  KMNL  cuts  AB  in  the  foci 
F  and  FI.  For  the  lines  KM,  ML,  and  LN  form  the  sides  of 


FIG.  91. 


a  triangle  circumscribed  about  the  hyperbola,  one  vertex  being 
infinitely  distant.  The  points  F  and  FI  are  conjugate  to  the  ideal 
vertex  and  are,  therefore,  joined  to  M  and  L  by  conjugate  lines. 
But  MF  and  LF  are  at  right  angles  to  each  other,  and  the  axis 
is  perpendicular  to  its  conjugate  chord  through  F.  Hence,  F 
is  a  focus.  Similarly,  FI  is  a  focus.  It  follows,  as  in  the  case  of  the 
ellipse,  that  F  and  FI  are  the  only  real  foci. 

3.  The  Parabola. — Let  S  be  the  vertex  of  the  parabola  (Fig.  92). 
Draw  the  axis  and  the  tangent  at  S.  Also,  a  second  tangent, 
meeting  the  first  at  N,  whose  point  of  contact  is  A.  The  lines 
SN  and  NA,  together  with  the  ideal  line,  form  a  circumscribed 
triangle  two  of  whose  vertices  are  ideal  points.  Any  point  on  the 
axis,  as  F,  is  conjugate  to  the  ideal  vertex  on  SN  and  is  joined 
to  the  other  two  vertices  by  conjugate  lines.  But  the  line  join- 
ing F  to  the  ideal  vertex  on  NA  is  parallel  to  NA.  If,  then, 
NF  is  drawn  perpendicular  to  NA,  its  conjugate  through  F  is 
perpendicular  to  it,  and  the  two  constitute  a  pair  of  normal  con- 


U20] 


PROPERTIES  OF  CONICS 


149 


jugate  rays  in  the  involution  set  up  about  F  by  the  parabola. 
The  axis  and  its  conjugate  chord  through  F  form  a  second  pair  of 
normal  conjugate  rays  in  the  same  involution.  Hence,  F  is  a 
focus. 

The  parabola  has  but  one  proper  focus.  It  is  sometimes 
convenient  to  regard  the  ideal  point  on  the  axis  as  a  secondary, 
or  improper,  focus. 


FIG.  92. 
Exercises 

1.  Show  that  the  circle  on  MN  as  diameter  (Fig.  90)  cuts  AB  in 
points  harmonically  separated  by  F,  F\. 

2.  Given  the  major  axis  AB  and  one  focus  of  an  ellipse,  or  of  a 
hyperbola,  construct  a  series  of  tangents  to  the  curve. 

3.  If  a  and  b  are  respectively  the  lengths  of  the  semi-major  axis  and 
the  semi-minor  axis  of  an  ellipse,  show  that  the  distance  between  the 
foci  is  2\/a2  -  b2. 

4.  If  2a  is  the  length  of  the  major  axis  of  a  hyperbola  and  26 
is  the  length  of  the  segment  of  a  tangent  at  one  vertex  contained  be- 
tween the  asymptotes,  show  that  the  distance  between  the  foci  is 
2\/aMr6~2. 

5.  Given  the  vertex  and  focus  of  a  parabola,  construct  a  series  of 
tangents  to  the  curve. 

120.  Directrices  and  Focal  Radii. — The  polar  line  of  a  focus 
with  respect  to  a  conic  is  called  a  directrix. 
A  circle  has  but  one  directrix,  the  ideal  line  in  its  plane. 


150  PROTECTIVE  GEOMETRY  [§121 

A  parabola  has  but  one  proper  directrix.  This  directrix 
cuts  the  axis  in  the  harmonic  conjugate  of  the  focus  with  respect 
to  the  two  curve  points  on  the  axis.  In  other  words: 

A  parabola  bisects  the  segment  of  its  axis  contained  between  the 
directrix  and  the  focus. 

An  ellipse,  or  a  hyperbola,  has  two  directrices  neither  of  which 
meets  the  curve  in  real  points. 

Lines  joining  any  point  on  a  conic  to  the  foci  are  called  focal 
radii,  or  focal  rays,  of  the  conic. 

From  each  point  of  a  circle  there  is  but  one  focal  radius,  namely, 
the  radius  of  the  circle  drawn  to  that  point. 

Likewise,  from  each  point  of  a  parabola  there  is  but  one  proper 
focal  radius;  but  it  is  often  convenient  to  consider  the  diameter 
through  the  point  as  a  second  focal  radius,  that  is,  as  the  radius 
drawn  to  the  ideal,  or  improper,  focus. 

There  are  two  focal  radii  from  each  point  of  an  ellipse,  or 
a  hyperbola. 

FOCAL  PROPERTIES  OF  CONICS 

121.  Fundamental  Theorem. — Theorem  XIV.— The  line  join- 
ing a  focus,  F,  of  a  conic  to  the  intersection  of  any  two  tangents 
to  the  conic  bisects  one  of  the  angles  between  the  focal  radii  from  F 
to  the  points  of  contact  of  the  tangents. 

Let  A  and  B  be  the  points  of  contact  of  tangents  drawn  from 
T,  and  let  /  be  the  directrix  corresponding  to  the  focus  F  (Fig. 
93).  If  AB  meets  the  directrix  in  P,  then  the  polar  line  of  P 
is  TF.  Hence  TF  and  PF  are  conjugate  lines  with  respect  to  the 
conic  and  consequently  perpendicular  to  each  other  (definition  of 
focus).  Since  the  points  A  and  B  are  separated  harmonically 
by  P  and  its  polar  line  TF,  we  have  a  harmonic  pencil  of  rays  about 
F  in  which  TF  and  PF  are  conjugate  to  each  other  and  hence 
bisect  the  interior  and  exterior  angles  between  the  other  two 
(Art.  27).  This  proves  the  theorem. 

122.  Consequences  of  Theorem  XIV. — 

1.  The  finite  segment  of  a  variable  tangent  to  a  conic,  contained 
between  two  fixed  tangents,  subtends  a  constant  angle  at  a  focus. 

For,  if  TA  and  TB  are  fixed  tangents  with  points  of  contact  at 
A  and  B,  respectively,  and  A\Bi  is  the  segment  of  a  variable  tan- 


§122] 


PROPERTIES  OF  CONICS 


151 


gent  contained  between  the  fixed  tangents,  and  touching  the  conic 
at  C,  then  AiF  bisects  the  angle  CFA,  and  B^F  bisects  the  angle 


FIG.  93. 


FIG.  94. 

CFB.  Consequently,  the  angle  AiFBi  is  half  the  angle  AFB  for 
all  positions  of  the  variable  tangent  AiB\.  But  AFB  is  a  fixed 
angle  (Fig.  94). 


152  PROJECTIVE  GEOMETRY  [§122 

As  a  direct  consequence  of  the  property  just  proved,  we  have: 

The  protectively  related  point-rows  in  which  any  two  tangents  to 

a  conic  are  cut  by  the  remaining  tangents  are  projected  from  a  focus 

of  the  conic  in  two  protectively  related  sheaves  of  rays  which  are 

equiangular. 

Exercises 

1.  Construct  a  series  of  tangents  to  a  conic,  having  given  one  focus 
and  three  tangents. 

2.  Given  one  focus,  two  tangents,  and  the  point  of  contact  on  one 
of  the  tangents;  construct  a  series  of  tangents  to  the  conic. 

3.  Show  that  the  circle  circumscribed  about  any  triangle,  whose 
sides  touch  a  parabola,  passes  through  the  focus  of  the  parabola 
(Lambert,  1761). 

4.  Show  that  the  four  circles  circumscribed  about  the  four  triangles 
formed  by  the  sides  of  a  complete  quadrilateral  pass  through  one  and 
the  same  point. 

5.  If  F  and  FI  are  the  foci  of  a  hyperbola,   and  any  tangent 
meets  the  asymptotes  in  the  points  A  and  B,  show  that  the  quad- 
rangle AFBFi  can  be  inscribed  in  a  circle. 

II.  Any  tangent  to  an  ellipse,  or  to  a  hyperbola,  bisects  one  of 
the  angles  between  the  focal  radii  drawn  to  its  point  of  contact. 

Let  T  be  the  point  of  contact  of  the  tangent,  and  F  and  FI  the 
foci  (Fig.  95).  Let  the  given  tangent  meet  the  tangent  at  the 
vertex  A  in  the  point  P.  The  lines  PF  and  PF\  are  bisectors  of 
.the  angles  TFA  and  TFtA,  respectively  (Art.  121).  Consequently, 
P  is  equidistant  from  the  sides  of  the  triangle  TFFi,  and  hence  on 
the  bisector  of  one  of  the  angles  at  T.  Therefore,  the  tangent  TP 
bisects  one  of  the  angles  between  the  focal  radii  TF  and  TF\,  as 
was  to  be  proved. 

The  circle  with  P  as  center  and  PA  as  radius  is  inscribed  in  the 
triangle  TFFi,  in  case  the  conic  is  a  hyperbola;  and  is  escribed 
to  the  triangle  TFFi,  in  case  the  conic  is  an  ellipse. 

III.  Any  tangent  to  a  parabola  bisects  one  of  the  angles  between 
the  focal  radius  to  the  point  of  contact  and  the  diameter  through  the 
point  of  contact. 

Thus,  Fig.  96,  the  given  tangent  meets  the  tangent  at  the  vertex 
A  in  the  point  P;  and  consequently  PF  bisects  the  angle  AFT 
(Art.  121)  and  is  perpendicular  to  PT  (Art.  119,  3).  The  triangle 


§122] 


PROPERTIES  OF  CONICS 


153 


KFT  is  therefore  isosceles,  the  angle  at  K  is  equal  to  the  angle  at 
T.     The  diameter  through  T  is  parallel  to  the  axis;  and  conse- 


K      A 


FIG.  96. 


quently  the  tangent  KT  bisects  one  of  the  angles  between  the 
focal  radius  FT  and  the  diameter  through  T. 


154  PROJECTIVE  GEOMETRY  [§122 

NOTE. — The  properties  proved  in  II  and  III  are  usually  called 
reflection  properties.  Consider  the  surface  generated  by  the 
revolution  of  a  conic  around  that  axis  upon  which  lie  the  real  foci. 
A  disturbance  originating  at  one  focus  and  travelling  in  straight 
lines  will  be  reflected  from  the  surface  and  proceed  in  straight 
lines  toward,  or  away  from,  the  other  focus.  Thus,  rays  of  light 
emanating  from  the  focus  of  a  paraboloid  of  revolution  will  be 
reflected  from  the  surface  in  parallel  rays.  In  the  case  of  an 
ellipsoid,  the  rays  will  converge  toward  the  other  focus;  in  the 
hyperboloid,  they  will  be  reflected  away  from  the  other  focus. 

IV.  The  ratio  of  the  distances  of  any  point  on  a  conic  from  a  focus 
and  from  the  corresponding  directrix  is  constant. 

In  Fig.  93,  let  F  be  a  focus  and  /  the  corresponding  directrix. 
If  AB  is  any  secant,  meeting  the  conic  in  A  and  B  and  the  direc- 
trix in  P,  then  FP  bisects  the  exterior  angle  at  F  of  the  triangle 
AFB.  Consequently, 

BP  _FB 
AP  ~  FA 

If  the  feet  of  the  perpendiculars  from  A  and  B  upon  the  directrix 
are  A2  and  B2,  respectively,  then 

BP  _  BB* 
AP  ~  AAZ' 
Hence, 

FB        FA 


BB2  ~~  AA2 
which  proves  the  proposition. 

NOTE.  —  This  property  is  often  used  in  analytic  geometry  to 
define  the  conic  sections.  The  constant  ratio  is  called  the  eccen- 
tricity of  the  conic.  The  property  itself  was  known  to  Pappus 
but  was  probably  first  used  as  a  definition  by  Boscovich  (1711- 
1787).  For  this  reason,  the  eccentricity  is  sometimes  called 
Boscovich's  ratio. 

Exercises 

1.  Show  that  the  eccentricity  of  a  parabola  is  unity  (Art.  120). 

2.  If  e  is  the  eccentricity,  show  that: 

•y/a2  _  5? 
e  =  --  '  for  the  ellipse; 


•  __ 

and  e  =  —          —  »  for  the  hyperbola. 


§122] 


PROPERTIES  OF  CONICS 


155 


What  is  the  significance  of  the  fact  that  the  common  limit  of  these 
expressions  asa=<»ise  =  l? 

Show,  also,  that  e  is  always  less  than  unity,  for  the  ellipse,  and 
always  greater  than  unity,  for  the  hyperbola. 

3.  Show  that  the  eccentricity  of  a  circle  is  zero.     How  will  a  conic 
change  in  form  as  its  eccentricity  increases  from  0  to  <»  ? 

4.  Given  a  focus,  its  corresponding  directrix,  and  the  eccentricity; 
construct  a  series  of  points  on  the  conic. 

5.  Given  a  focus,  its  corresponding  directrix,  and  one  point;  con- 
struct a  number  of  points  on  the  conic. 

6.  Show  that,  for  the  ellipse  or  the  hyperbola,  the  distances  from  the 
center  to  a  focus  and  from  the  center  to  the  corresponding  directrix 
are  ae  and  a/e,  respectively. 

7.  Given  the  foci  and  a  point  P  on  an  ellipse,  or  a  hyperbola,  con- 
struct the  tangent  at  P. 

8.  Given  the  focus,  the  axis,  and  a  point  P  on  a  parabola;  construct 
the  tangent  at  P. 

9.  Given  a  right  circular  cylinder,  show  that  the  eccentricity  of  an 
ellipse,  cut  from  the  cylinder  by  a  plane  passing  through  a  diameter 
of  a  circular  section  and  making  an  angle  x  with  this  section,  is  e  = 
sin  x. 


FIG.  97. 


V.  The  sum  of  the  focal  radii  to  any  point  on  an  ellipse  is  constant. 
The  difference  of  the  focal  radii  to-  any  point  on  a  hyperbola  is 
constant. 

For,  in  Fig.  97,  let  r  and  n  be  the  focal  radii  to  any  point  A, 
and  d  and  d\  the  distances  to  the  corresponding  directrices.  Since 


156  PROJECTIVE  GEOMETRY  [§122 

an  ellipse,  or  a  hj'-perbola,  is  symmetric  with  respect  to  either 
axis,  we  have 


Therefore,  also, 

r  ±  n  =  e(d  ± 


But  for  the  ellipse  d  +  d\  (for  the  hyperbola  d  —  d\)  is  constant 
for  all  positions  of  the  curve-point  A,  this  number  expressing  the 

distance  between  the  directrices.  If 
A  is  taken  at  a  vertex,  we  see  that 
the  sum  of  the  lengths  of  the  focal 
radii  to  any  point  on  an  ellipse  is 
equal  to  the  length  of  the  major  axis; 
and  that  the  difference  of  the  lengths 
of  the  focal  radii  to  any  point  on  a 
hyperbola  is  equal  to  the  length  of  the 
major  axis. 

pIG   gg  VI.  The  feet  of  the  perpendiculars 

from  the  foci   of  an   ellipse,   or   of  a 

hyperbola,  upon  a  variable  tangent  lie  upon  a  circle  which  is  con- 
centric with  the  curve  and  whose  diameter  is  equal  to  the  major  axis 
of  the  curve. 

Let  F  and  FI  be  the  foci  and  A  the  point  of  contact  of  a  variable 
tangent  (Fig.  98).  If  the  perpendicular  from  FI  on  the  tangent 
at  A  meets  the  focal  radius  FA  in  G,  then  AG  is  equal  to  AFi,  since 
the  tangent  bisects  the  angle  GAF\.  Hence  FG  =  the  length  of 
the  major  axis  =  2a.  If  M  is  the  center  of  the  curve  and  N  is 
the  foot  of  the  perpendicular  upon  the  tangent,  then  MN  =  Y^FG 
—  a.  Consequently,  the  locus  of  N  is  a  circle  whose  radius  is  a. 
This  circle  is  called  the  major  auxiliary  circle. 

A  figure  should  be  drawn  and  the  proposition  proved  for  the 
hyperbola. 

In  the  case  of  the  parabola,  we  have  (Art.  119,  3): 
The  locus  of  the  foot  of  a  perpendicular  from  the  focus  of  a  parabola 
upon  a  variable  tangent  is  the  tangent  at  the  vertex  of  the  curve. 


§122]  PROPERTIES  OF  CONICS  157 

Exercises 

1.  Given  the  foci  and  the  length  of  the  major  axis  of  an  ellipse  or 
of  a  hyperbola,  construct  the  tangents  from  an  exterior  point  to  the 
curve  and  their  points  of  contact. 

Suggestions. — Let  F  and  FI  be  the  foci  and  P  the  point  from  which 
the  tangents  are  to  be  drawn.  With  P  as  center  draw  a  circle  through 
the  nearer  focus,  and  with  the  other  focus  as  center  and  the  major 
axis  as  radius  draw  an  arc  cutting  the  first  circle  in  the  points  D  and  DI. 
The  perpendicular  bisectors  of  the  segments  FD  and  FDi  are  the  re- 
quired tangents,  and  the  points  in  which  these  bisectors  meet  the 
lines  joining  D  and  Di  to  the  farther  focus  are  the  required  points 
of  contact. 

2.  Given  the  focus  and  the  directrix  of  a  parabola,  construct  the 
tangents  from  an  exterior  point  to  the  curve  and  their  points  of 
contact. 

Suggestions. — Let  F  be  the  focus,  /  the  directrix,  and  P  the  point 
from  which  the  tangents  are  to  be  drawn.  Draw  the  circle  with  PF 
as  radius  and  P  as  center,  meeting  the  directrix  in  D  and  D\.  The 
required  tangents  are  the  perpendicular  bisectors  of  the  segments  FD 
and  FDi,  and  the  points  in  which  these  bisectors  meet  the  diameters 
through  D  and  DI  are  the  respective  points  of  contact. 

3.  For  an  ellipse  or  a  hyperbola,  the  four  focal  radii  drawn  from 
the  extremities  of  any  chord  touch  a  circle  whose  center  is  the  pole  of 
the  chord.     State  and  prove  the  corresponding  property  for  the  para- 
bola (Chasles,  1830). 

4.  Prove  that  the  subnormal  of  a  parabola  is  constant  in  length. 

6.  Show  that  the  locus  of  the  center  of  a  circle  touching  two  fixed 
circles  is  a  conic.  Discuss  the  problem  for  various  positions  of  the 
fixed  circles. 

6.  Construct  a  series  of  points  of  a  conic  having  given  one  focus, 
the  corresponding  directrix,  and  one  tangent. 

7.  If  an  ellipse  and  a  hyperbola  are  confocal,  show  that  they  inter- 
sect at  right  angles. 

VII.  The  product  of  the  perpendiculars  from  the  foci  of  an  ellipse, 
or  of  a  hyperbola,  upon  a  variable  tangent  is  constant. 

The  feet  of  the  perpendiculars  lie  upon  the  major  auxiliary 
circle  by  the  preceding  property.  If  the  perpendiculars  meet  the 
auxiliary  circle  again  in  the  points  N  and  MI  (Fig.  99),  then  NM 
and  NiMi  are  diameters  of  this  circle.  Hence  we  have,  F^Ni-FM 
=  FiNyFiN  =  AFvFiB  =  (AO  -  OF}(AO  +  OF)  =  a2  -  OF2  = 
62  (Art.  119,  exercise  3). 


158  PROJECTIVE  GEOMETRY  [§122 

The  student  should  prove  the  proposition  for  the  hyperbola, 
making  use  of  Art.  119,  exercise  4. 

VIII.  The  locus  of  the  vertex  of  a  right  angle  whose  sides  touch  an 
ellipse  or  a  hyperbola  is  a  circle  concentric  with  the  conic. 

Thus,  if  R  is  the  vertex  of  a  right  angle  whose  sides  touch  the 
ellipse  (Fig.  99),  and  S  and  T  are  the  feet  of  perpendiculars  from 
the  foci  upon  one  side,  then 

M    K  FiSFT  =  NiR-MR  =  b2  =  RK\ 

where  RK  is  the  length  of  the  tangent 
from  R  to  the  major  auxiliary  circle. 
Hence 


=  OK2  +  RK*  =  a2  +  62. 

The  locus  of  R  is,  therefore,  the  circle 
whose  center  is  0  and  whose  radius  is 
the  constant  distance  OR.  This 
circle  is  called  the  director  circle  for 
the  conic  in  question. 
The  student  should  construct  a  figure  and  prove  the  proposition 
for  the  hyperbola.  The  radius  of  the  director  circle  for  the  hyper- 
bola is  \a2  —  bz,  and  consequently  the  hyperbola  has  a  real 
director  circle  only  when  b  <  a;  that  is,  only  when  the  angle 
between  the  asymptotes,  within  which  the  curve  lies,  is  acute. 

Exercises 

1.  Prove  that  the  locus  of  the  vertex  of  a  right  angle  whose  sides 
touch  a  parabola  is  the  directrix. 

2.  Show  that  the  directrices  of  all  parabolas  which  can  be  inscribed 
in  a  given  triangle  pass  through  the  orthocenter  of  the  triangle  and 
the  foci  lie  on  the  circumscribed  circle. 

3.  If  perpendiculars  are  dropped  from  any  point  of  a  circle  upon  the 
sides  of  an  inscribed  triangle,  prove  that  the  feet  of  these  perpendicu- 
lars lie  upon  a  straight  line  which  is  the  tangent  at  the  vertex  of  one 
of  the  parabolas  inscribed  in  the  triangle  (Simson  line,  or  Wallace  line). 

4.  If  P  is  a  point  outside  an  ellipse  or  a  hyperbola,  show  that  the 
lines  joining  P  to  the  foci  are  equally  inclined  to  the  tangents  drawn 
from  P. 

6.  A  conic  is  inscribed  in  a  given  triangle  ABC  and  F  is  one  focus. 
Construct  the  other  focus. 


CHAPTER  XIV 

PROJECTIVELY  RELATED   PRIMITIVE   FORMS   OF   THE 
SECOND  KIND 

123.  Primitive  Forms  of  the  Second  Kind.    Incident  Elements. 

—The  field  of  points,  the  field  of  rays,  the  bundle  of  rays,  and  the 
bundle  of  planes  constitute  a  group  of  primitive  forms  such  that 
each  form  can  be  derived  from  one  of  the  others  by  projection  or 
section.  Thus,  the  projector  of  a  field  of  rays  is  a  bundle  of  planes. 
Again,  the  dual,  or  reciprocal,  of  one  of  these  forms  is  a  form  be- 
longing to  the  same  group.  The  forms  above  enumerated  are 
called  primitive  forms  of  the  second  kind.  Primitive  forms  of  the 
second  kind  are  two-dimensional  forms  (Art.  15). 

A  field  of  points  and  a  field  of  rays  can  be  superposed,  thus  form- 
ing a  plane ;  and  a  bundle  of  rays  and  a  bundle  of  planes  can  be 
superposed,  forming  a  bundle.  The  plane  and  the  bundle  are 
reciprocal,  or  dual,  figures  in  space;  and  the  one  is  a  section  of  the 
other.  A  plane  contains  all  the  points  and  lines  lying  on  it;  and 
a  bundle  is  composed  of  all  the  lines  and  planes  passing  through  its 
vertex. 

If,  in  a  plane,  the  point  A  lies  on  the  line  a,  the  two  elements 
A ,  a  are  said  to  be  incident  elements.  Similarly,  in  a  bundle,  if  the 
line  a  lies  in  the  plane  a,  the  two  elements  a,  a  are  incident 
elements.  Thus,  in  a  plane,  a  point  is  incident  to  all  the  lines 
passing  through  it,  and  a  line  is  incident  to  all  the  points  lying  on  it. 
In  a  bundle,  a  line  is  incident  to  all  the  planes  through  it,  and  a 
plane  is  incident  to  all  the  lines  of  the  bundle  lying  in  it  (cf.  Art.  14). 

124.  Perspective  Position  of  Planes  and  Bundles. — A  plane  and 
a  bundle  are  in  perspective  position  if  the  plane  is  a  section  of  the 
bundle,  each  element  of  the  plane  then  lies  on  its  corresponding 
element  of  the  bundle. 

Two  planes  are  in  perspective  position  if  each  is  a  section  of 
the  same  bundle.  Corresponding  elements  in  the  planes  then 
lie  on  the  same  element  of  the  bundle. 

Two  bundles  are  in  perspective  position  if  each  is  a  projector  of 

159 


160  PROJECTIVE  GEOMETRY 

the  same  plane.  Corresponding  elements  in  the  bundles  then 
have  in  common  the  same  element  of  the  plane. 

A  series  of  planes  and  bundles,  each  of  which  is  in  perspective 
position  with  the  one  following  it,  constitutes  a  chain  of  perspec- 
tivity.  The  first  and  the  last  of  a  chain  of  perspectivity  are  corre- 
lated, element  to  element,  but  are  not,  in  general,  in  perspective 
position. 

If  two  planes,  or  two  bundles,  are  in  perspective  position,  or  are 
correlated  by  means  of  a  chain  of  perspectivity  connecting  them, 
corresponding  elements  are  always  of  the  same  kind;  that  is,  point 
corresponds  to  point  and  line  to  line,  or  line  to  line  and  plane  to 
plane. 

125.  Orthogonally  Correlated  Bundles. — Two  bundles  can  be 
correlated  so  that  to  any  element  of  either  (line  or  plane)  corre- 
sponds that  element  of  the  other  (plane  or  line)  which  is  perpen- 
dicular to  it.     If  S  and  Si  are  the  centers  of  the  bundles,  then  two 
lines  a  and  b  of  S  correspond  to  the  planes  «i  and  /3i  of  S\  which 
are  respectively  perpendicular  to  a  and  b.    The  plane  (ao)  of  S 
corresponds  to  the  line  («]/3i)  of  Si;  and  it  is  clear  that  (ab)  is 
perpendicular  to  («i/3i).     Two  bundles  correlated  in  this  manner 
are  said  to  be  orthogonally  correlated. 

If  we  cut  each  of  two  orthogonally  correlated  bundles  by  planes, 
the  sections  so  obtained  are  correlated,  element  to  element,  but 
are  neither  in  perspective  position  nor  are  they  the  first  and  last 
of  a  chain  of  perspectivity.  In  fact  they  are  so  correlated  that 
point  corresponds  to  line  and  line  to  point. 

126.  Definition  of  Projective  Relationship. — To  avoid  circum- 
locution, the  plane  and  the  bundle  will  be  called  forms  of  the 
second  kind. 

Two  correlated  forms  of  the  second  kind  are  projectively  related  if 
incident  elements  in  either  always  correspond  to  incident  elements  in 
the  other. 

Thus  two  correlated  forms  of  the  second  kind  are  projectively 
related  if  they  are  in  perspective  position,  or  if  they  are  the  first  and 
last  of  a  chain  of  perspectivity.  Again,  two  bundles  are  pro- 
jectively related  if  they  are  orthogonally  correlated;  two  planes  are 
projectively  related  if  they  are  sections  of  orthogonally  correlated 
bundles. 


§127]      PRIMITIVE  FORMS  OF  THE  SECOND  KIND       161 

127.  Definition  of  Collineation. — //  two  forms  of  the  second  kind 
are  protectively  related  and  corresponding  elements  are  of  like  kinds, 
the  forms  are  said  to  be  collinearly  related  and  the  projective  relation- 
ship is  called  a  collineation. 

Thus,  if  two  forms  are  in  perspective  position,  or  are  connected 
by  a  chain  of  perspectivity,  they  are  collinearly  related. 

128.  Definition  of  Duality. — //  two  forms  of  the  second  kind 
are  protectively  related  and  corresponding  elements  are  of  unlike 
kinds,  the  forms  are  said  to  be  reciprocally  related  and  the  projective 
relationship  is  called  a  duality. 

Thus  if  two  bundles  are  orthogonally  correlated,  they  are 
reciprocally  related.  The  plane  sections  of  two  orthogonally 
correlated  bundles  are  reciprocally  related  planes. 

129.  Consequences  of  the  Foregoing  Definitions. — 

1.  Two  forms  of  the  second  kind,  each  collinearly  related  to  a 
third  form,  are  collinearly  related  to  each  other. 

2.  Two  forms  of  the  second  kind,  each  reciprocally  related  to  a 
third  form,  are  collinearly  related  to  each  other. 

3.  Two  forms  of  the  second  kind,  one  reciprocally  related  and  the 
other  collinearly  related  to  a  third  form,  are  reciprocally  related  to 
each  other. 

4.  If  two  forms  of  the  second  kind  are  projectively  related,  any 
primitive  form  of  the  first  kind  in  either  corresponds  to  a  primitive 
form  of  the  first  kind  in  the  other. 

Thus,  any  point-row  in  one  of  two  collinearly  related  planes 
corresponds  to  a  point-row  in  the  other.  If  the  planes  are  re- 
ciprocally related,  any  point-row  in  one  corresponds  to  a  sheaf 
of  rays  in  the  other.  In  reciprocally  related  bundles,  a  sheaf 
of  rays  in  one  corresponds  to  a  sheaf  of  planes  in  the  other. 

Primitive  forms  of  the  first  kind  that  correspond  to  each  other 
in  projectively  related  forms  of  the  second  kind  are  called  corre- 
sponding primitive  forms  of  the  first  kind. 

130.  Fundamental  Theorem. — Theorem  XV. — //  two  forms  of 
the  second  kind  are  projectively  related,  then  corresponding  primitive 
forms  of  the  first  kind  are  projectively  related  to  each  other. 

Suppose  that  a  and  ai  are  collinearly  related  planes,  and  that 
u  and  u\  are  corresponding  point-rows  (Fig.  100).    We  are  to 
show  that,  in  the  collineation  existing  between  the  planes,  any 
11 


162 


PROJECTIVE   GEOMETRY 


[§130 


four  harmonic  points  of  u  correspond  to  four  harmonic  points 
of  MI.  Let  A  BCD  be  any  harmonic  range  on  u  defined  by  a 
complete  quadrangle  KLMN  in  a,  and  let  corresponding  points 
in  at  be  denoted  by  subscripts,  so  that  A\  corresponds  to  A,  B\ 
to  B,  and  so  on.  Since  the  planes  are  projectively  related,  col- 
linear  points  in  one  correspond  to  collinear  points  in  the  other, 


FIG.  100. 

and  concurrent  lines  to  concurrent  lines.  It  follows  at  once 
that  AiBiCiDi  is  a  harmonic  range  on  MI  defined  by  the  complete 
quadrangle  KiLiMiNi.  Since  ABCD  is  any  harmonic  range  on 
u,  and  u  is  any  point-row  in  a,  it  follows  that  any  two  correspond- 
ing point-rows  are  projectively  related. 

If  N  and  NI  are  the  centers  of  any  two  corresponding  sheaves 
of  rays,  then 

N  7\  M  A  Ui  ~/\  NI, 

where  u  and  u\  are  corresponding  sections  of  N  and  A'i  respectively. 
Hence,  N  A  NI. 

If  «  and  <*i  are  reciprocally  related  planes,  a  harmonic  range 
and  its  defining  quadrangle  in  «  corresponds  to  a  pencil  of  rays 
and  a  quadrilateral  in  at.  Since,  in  the  duality  existing  between 
the  planes,  collinear  points  correspond  to  concurrent  lines,  and 
concurrent  lines  to  collinear  points,  it  is  an  easy  matter  to  show 
that  the  corresponding  pencil  is  harmonic,  and  hence  that  corre- 
sponding primitive  forms  of  the  first  kind  are  projectively  related. 
The  student  should  complete  the  proof. 


§131]     PRIMITIVE  FORMS  OF  THE  SECOND  KIND       163 


If  the  projectively  related  forms  are  bundles,  each  can  be  cut 
by  a  plane,  and  the  proof  of  the  theorem  thus  reduced  to  the 
cases  already  considered. 

Finally,  if  a  plane  is  projectively  related  to  a  bundle,  the 
theorem  still  holds  for  these  forms,  since  the  given  plane  is 
projectively  related  to  any  plane  section  of  the  given  bundle. 

131.  Determination  of  Projective  Relationship. — A  protective 
relationship  between  two  forms  of  the  second  kind  is  uniquely  deter- 
mined by  choosing  any  four  elements  of  like  kind  in  one,  no  three  of 
which  belong  to  the  same  primitive  form  of  the  first  kind,  to  correspond 


FIG.  101. 

to  four  arbitrarily  chosen  elements  of  like  kind  in  the  other,  no  three 
of  which  belong  to  any  primitive  form  of  the  first  kind. 

For  example,  a  collineation  is  established  between  two  planes 
when  the  vertices  of  any  quadrangle  A  BCD  in  one  are  chosen  to 
correspond  respectively  to  the  vertices  AiBiCiDi  of  any  quad- 
rangle in  the  other  (Fig.  101).  For  then 

A(BCD )  7^A1(B1D1C1 ) 

and 

B(ACD )  X  SiCAiCiDi ), 

the  three  pairs  of  corresponding  rays,  in  each  case,  being  just 
sufficient  to  determine  the  projectivity  between  the  corresponding 
sheaves  of  rays.  If  P  is  any  point  in  a,  we  can  construct  the  rays 
AiPi  and  B\Pi  in  «i  which  correspond  respectively  to  the  rays 
AP  and  BP  in  a.  Hence,  PI  is  uniquely  determined. 

If  p  is  any  line  in  «  and  P  and  Q  are  any  two  points  on  p,  we 
can  construct  the  points  PI  and  Qi  in  a\;  and  hence  the  line  p\ 
is  uniquely  determined.  The  collineation  is  therefore  completely 
determined. 


164  PROJECTIVE  GEOMETRY  [§131 

A  collineation  is  also  completely  determined  between  two 
planes  by  choosing  the  sides  of  any  complete  quadrilateral  in 
one  to  correspond  respectively  to  the  sides  of  any  complete 
quadrilateral  in  the  other. 

A  duality  is  completely  determined  between  two  planes  by 
choosing  the  vertices  of  any  complete  quadrangle  in  one  to 
correspond  respectively  to  the  sides  of  a  complete  quadrilateral 
in  the  other.  The  student  should  construct  a  figure  and  supply 
the  details  of  the  proof. 

Since  protectively  related  bundles  can  be  cut  in  protectively 
related  planes,  it  follows  that:  A  collineation  is  determined 
between  two  bundles  if  the  edges  (faces)  of  any  complete  4- 
edge  (4-face)  in  one  are  chosen  to  correspond  respectively  to 
the  edges  (faces)  of  any  complete  4-edge  (4-face)  in  the  other; 
and  a  duality  is  determined  between  two  bundles  if  the  edges 
(faces)  of  a  complete  4-edge  (4-face)  in  one  are  chosen  to  corre- 
spond respectively  to  the  faces  (edges)  of  any  complete  4-face 
(4-edge)  in  the  other. 

Exercises 

1.  If  two  planes  are  collinearly  related,  prove  that  a  conic  in  either 
corresponds  to  a  conic  in  the  other.     If  the  planes  are  reciprocally 
related,  what  corresponds  to  a  conic  in  one? 

2.  If  a  plane  is  protectively  related  to  a  bundle,  what  corresponds 
to  a  conic  in  the  plane?     There  are  two  cases. 

3.  If  two  bundles  are  orthogonally  correlated,  show  that  corre- 
sponding elements  intersect  upon  a  sphere  whose  center  bisects  the 
line  joining  the  centers  of  the  given  bundles. 

4.  If  two  bundles  are  reciprocally  related,  show  that  the  surface 
upon  which  lie  the  points  of  intersection  of  corresponding  elements  is, 
in  general,  cut  by  any  plane  of  either  bundle  in  a  conic.     Examine 
the  section  of  this  surface  by  that  plane  of  either  bundle  which  cor- 
responds to  the  line  joining  the  centers  of  the  bundles. 

6.  A  plane  is  divided  into  squares  by  lines  drawn  parallel  to  two 
perpendicular  axes  as  in  analytic  geometry.  If  the  origin,  the  ideal 
point  on  each  axis,  and  the  point  whose  coordinates  are  (1,  1)  are  made 
to  correspond  to  the  vertices  of  any  complete  quadrangle  in  a  second 
plane,  show  how  to  divide  the  second  plane  into  quadrangles  corre- 
sponding to  the  squares  in  the  first  plane.  If  the  coordinates  of  a 
point  in  the  first  plane  are  any  two  rational  numbers,  construct  the 


§133]      PRIMITIVE  FORMS  OF  THE  SECOND  KIND       165 

corresponding  point  in  the  second  plane.     Can  the  construction  be 
effected  if  the  coordinates  are  irrational  numbers? 

6.  Two  bundles  are  orthogonally  correlated  and  concentric.  Show 
that  the  duality  determined  upon  a  plane  section  of  the  bundles  is 
involutoric.  If  the  bundles  are  not  concentric,  will  the  duality  on  a 
plane  section  be  involutoric?  Show  that,  in  the  second  case,  the 
locus  of  points  which  lie  on  their  corresponding  lines  is,  in  general,  a 
circle. 

132.  Projectivities  in  a  Form  of  Second  Kind. — If  two  forms  of 
the  second  kind  are  projectively  related,  we  shall  say  that  a  pro- 
jectivity  exists  between  them,  and  we  shall  regard  this  projectivity 
as  a  transformation  which  transforms  one  form  into  the  other. 
If  the  forms  can  be  superposed,  that  is,  if  they  are  both  planes  or 
both  bundles,  the  projectivity  transforms  their  common  base  into 
itself.     Thus,  two  collinearly  related  and  superposed  planes  con- 
stitute a  collineation  which  transforms  their  common  plane  into 
itself,  replacing  each  point  by  another  (or  in  particular  cases  by 
the  same)  point,  and,  in  general,  each  line  by  another  line.    A 
duality  in  a  plane  results  from  superposing  two  reciprocally  related 
planes,  and  replaces  each  point  of  the  common  plane  by  a  line  of 
that  plane,  and  each  line  by  a  point. 

In  the  succeeding  articles  we  shall  study,  in  some  detail,  various 
types  of  collineations. 

133.  The  Perspectivity. — If  two  collinearly  related  planes  are 
in  perspective  position,  they  must  have  their  line  of  intersection 
as  a  self-corresponding  line.     For  every  point  of  this  line,  consid- 
ered as  a  point  in  either  plane,  coincides  with  its  corresponding 
point  in  the  other  plane. 

Likewise,  the  line  joining  the  centers  of  two  bundles  in  perspec- 
tive position  is  a  self-corresponding  line.  For  every  plane  through 
this  line,  considered  as  a  plane  of  either  bundle,  coincides  with  its 
corresponding  plane  of  the  other  bundle. 

Conversely : 


//  two  planes  are  collinearly  re- 
lated and  have  a  self-correspond- 
ing line,  they  are  in  perspective 
position,  or  else  they  are  super- 
posed and  have  in  common  a 
sheaf  of  rays. 


If  two  bundles  are  collinearly 
related  and  have  a  self-correspond- 
ing ray,  they  are  in  perspective 
position,  or  else  they  are  super- 
posed and  have  in  common  a 
sheaf  of  rays. 


166 


PROJECTIVE  GEOMETRY 


[§134 


On  the  left,  if  ABC  and  AiBiCi  are  an>  two  corresponding  tri- 
angles and  u  is  the  self-corresponding  line,  then  corresponding 
sides  of  these  triangles  must  meet  on  u.  The  triangles  are  then 
in  perspective  position  (Art.  17),  and  lines  joining  corresponding 
vertices  intersect  in  a  point  U.  Any  fourth  point  D  is  joined  to 
its  corresponding  point  DI  by  a  line  through  U,  since  the  quad- 
rangles ABCD  and  A\BiC\Di  are  in  perspective  position  (Art.  20). 
The  planes  are  then  either  sections  of  the  same  bundle  U,  and  con- 
sequently in  perspective  position,  or  else  they  are  superposed  and 
have  in  common  the  sheaf  of  rays  U.  In  the  latter  case  we  shall 
say  they  constitute  a  perspectivity  in  their  common  plane. 

The  theorem  on  the  right  is  the  space-dual  of  the  theorem  on 
the  left.  The  bundles  are  then  either  the  projectors  of  the  same 


FIG.  102. 

plane,  and  consequently  in  perspective  position,  or  else  they  are 
superposed  and  have  in  common  a  sheaf  of  rays.  In  the  latter  case 
we  shall  say  they  constitute  a  perspectivity  in  their  common 
bundle. 

A  perspectivity  in  a  plane  has  a  self-corresponding  point-row, 
called  the  axis  of  perspectivity,  and  a  self-corresponding  sheaf  of 
rays  whose  center  is  called  the  center  of  perspectivity  (cf.  Art.  19). 

A  perspectivity  in  a  bundle  has  a  self-corresponding  sheaf  of 
planes  and  a  self-corresponding  sheaf  of  rays. 

134.  Construction  of  a  Perspectivity  in  a  Plane. — Given  the  axis 
and  the  center  of  a  perspectivity  and  one  pair  of  corresponding  points, 


§135]     PRIMITIVE  FORMS  OF  THE  SECOND  KIND        167 


or  one  pair  of  corresponding  rays,  to  construct  as  many  pairs  of  cor- 
responding elements  as  may  be  desired. 

Let  u  and  U  be  respectively  the  given  axis  and  the  given  center, 
and  A,  A i  the  given, pair  of  corresponding  points  (Fig.  102).  If 
B  is  any  point  in  the  plane,  the  lines  AB  and  A\B^_  must  meet  on 
u,  and  the  line  BB\  must  pass  through  U.  Hence  the  point  B\, 
corresponding  to  B,  is  immediately  constructed. 

If  p  is  any  ray  in  the  plane  and  C  is  any  point  on  p,  we  can  con- 
struct Ci,  corresponding  to  C;  and  then  p\  must  pass  through  d 
and  meet  p  on  the  axis  u. 

Exercises 

1.  Work  out  the  details  of  the  proof  for  the  theorem  on.  the  right 
in  Art.  133. 

2.  Construct  a  number  of  corresponding  elements  having  given  the 
axis  and  the  center  of  a  perspectivity  in  the  plane  and  one  pair  of 
corresponding  rays. 

3.  Given  one  pair  of  corresponding  points,  construct  pairs  of  cor- 
responding elements  in  a  perspectivity:  (a)  when  the  axis  is  the  ideal 
line  in  the  plane  and  the  center  is  an  actual  point;  (b)  when  the  axis 
is  an  actual  line  and  the  center  is  ideal;  (c)  when  both  axis  and  center 
are  ideal. 

135.  The  Invariant  of  a  Perspectivity  in  a  Plane. — Let  u  and  U 
be  the  axis  and  the  center  of  a 
perspectivity,  and  A,  AI  any 
pair  of  corresponding  points  (Fig. 
103).  If  the  line  UAA±  meets 
the  axis  in  0,  we  can  prove 
that  the  cross-ratio  (Art.  28) 
of  the  range  UAOAi  is  con- 
stant for  all  positions  of  A.  If 
B  and,  BI  are  a  pair  of  cor- 
responding points,  then  the 
ranges  UAOAl  and  UBMB, 

are  sections  of  the  same  pencil  whose  center  is  a  point  Q  on 
the  axis.  It  is  easy  to  show  that: 

UA       UAl      sin  UQA      sin  UQA, 

(1)       -TTV  -f-  ~iT7\  =  ~ — T7vr7  ~="  ~ — A  r\r\  (c/.  Art.  32,  exercise  4). 
AO       AiO      sin  AQO       sm  AiQO  w 


168  PROJECTIVE  GEOMETRY  [§136 

But  the  cross- ratio  of  the  range  UBMBi  is  equal  to  the  same 
constant,  since  this  constant  depends  only  upon  the  angles  at  Q : 
The  right  hand  member  of  (1)  is  the  cross-ratio  of  the  pencil 
Q(UAOAi).  This  cross-ratio  is  clearly  the  same  for  any  other 
pencil  of  which  UAOAi  is  a  section.  We  conclude,  therefore, 
that  the  cross-ratio  is  the  same  for  all  ranges  formed  like 
UAOAi;  and  that  this  constant  is  equal  to  the  cross-ratio  of  all 
pencils  formed  like  Q(UAOAi).  This  constant  is  called  the 
invariant  of  the  perspectivity. 

136.  The  Harmonic  Perspectivity,  or  Involution,  in  a  Plane. — 
The  perspectivity  whose  invariant  is    —1  is  called  a  harmonic 
perspectivity,  or  an  involution.     In  this  case,  the  ranges  UAOAi, 
UBMBi,   etc.,   are   harmonic    (Art.   28).    The   lines   ABi    and 
AiB  (Fig.  103)  meet  on  the  axis  in  a  point  T,  and  consequently  A 
and  A  i  correspond  doubly.     In  the  same  way,  the  points  of  any 
pair  correspond  to  each  other  doubly.     There  is  an  involution 
along  each  line  through  the  center  whose  foci  are  the  center  and 
the  point  in  which  the  line  meets  the  axis  u;  and  there  is  an  invo- 
lution about  each  point  of  the  axis  whose  focal  rays  are  the  axis  and 
the  ray  joining  the  point  to  the  center   U.    A  harmonic  per- 
spectivity is  thus  involutoric. 

The  collineation  in  a  plane  determined  by  choosing  the  vertices 
A,  AI,  B,  BL  of  a  quadrangle  to  correspond  respectively  to  the 
points  AI,  A,  BI,  B  is  an  involution. 

137.  Limiting  Lines  in    a   Collineation. — If   two   planes   are 
collinearly  related,  the  ideal  line  in  either  corresponds,  in  general, 
to  an  actual  line  in  the  other.     The  actual  line  in  either  plane 
which  corresponds  to  the  ideal  line  in  the  other  is  called  the  limiting 
line  in  its  plane. 

With  the  aid  of  the  limiting  lines  we  can  place  two  collinearly 
related  planes  in  perspective  position.  Thus  (Fig.  104),  let  a  and 
ai  be  collinearly  related  planes,  and  suppose  c  and  di  are  respec- 
tively the  limiting  lines  in  a  and  «i.  To  the  ideal  point  J  on  c 
corresponds  the  ideal  point  Ji  on  di,  and  hence  to  the  sheaf  of  rays 
whose  center  is  J  corresponds  the  sheaf  of  rays  whose  center  is  J\. 
Let  a,  b  be  a  pair  of  parallel  rays  in  a  not  belonging  to  the  sheaf  /. 
The  corresponding  pair  ai,  61  in  ai  must  intersect  on  di.  Suppose 
the  rays  a,  b  determine  a  segment  r  units  in  length  upon  c.  In  the 


§137]     PRIMITIVE  FORMS  OF  THE  SECOND  KIND       169 


plane  a\  select  the  two  rays  e\  and/i  of  the  sheaf  J\  upon  which  the 
rays  a\,  bi  determine  segments  r  units  in  length.  We  now  have  a 
quadrilateral  aiMi/i  in  a\  whose  vertices  A\BiC\D\  correspond  to 
the  vertices  of  a  parallelogram  A  BCD  in  a  each  to  each.  The 
two  planes  can  be  placed  so  that  the  points  A,  B,  J  coincide  with 
their  corresponding  points  AI,  BI,  Ji.  The  projectively  related 


FIG.  104. 

point-rows  e  and  e\  will  then  coincide  throughout  and  the  planes 
are  consequently  in  perspective  position. 

The  point-rows  /  and  f\  can  also  be  placed  in  coincidence ;  hence 
two  collinearly  related  planes  whose  ideal  lines  do  not  correspond 
to  each  other  can  always  be  placed  in  perspective  position  in  at 
least  two  ways. 

Exercises 

1.  Given  the  center  and  the  axis  of  a  perspectivity  in  a  plane  and 
one  pair  of  corresponding  points,  P  and  Pi.     If  P  is  made  to  describe 
any  conic,  construct  the  conic  described  by  Pt. 

2.  Construct  the  limiting  lines  of  a  perspectivity  in  a  plane.     Show 
that    the   limiting   lines    are   necessarily    parallel   to    the    axis    of 
perspectivity. 

3.  If   the   points   ABCD   of   a   plane  correspond  to   the  points 
BADC,  each  to  each ,  show  that  the  collineation  so  determined  is  an 
involution. 

4.  Two  collinearly  related  planes  a  and  ai  are  in  perspective  posi- 
tion.    If  a  is  rotated  around  their  common  line  M,  without  disturbing 
the  relative  position  of  its  elements,  show  that  the  center  of  the  bundle 
of  which  a  and  ai  are  sections  will  describe  a  circle  whose  center  is  in 


170  PROJECTIVE  GEOMETRY  [§138 

ai  and  corresponds  to  an  ideal  point  in  a,  the  plane  of  the  circle  being 
perpendicular  to  ju. 

5.  Given  two  collinearly  related  planes  whose  ideal  lines  do  not 
correspond  to  each  other,  find  a  sheaf  of  rays  in  one  to  which  corre- 
sponds a  congruent  sheaf  of  rays  in  the  other. 

Suggestions. — Choose  two  right  angles  in  one  plane  whose  sides 
a,  b  and  c,  d  do  not  form  a  parallelogram.  Let  A,  B,  C,  D  be  the  ideal 
points  on  a,  b,  c,  d  respectively,  and  A\,  Bi,  C\,  Di.be  the  corresponding 
points  on  the  limiting  line  in  the  other  plane.  Draw  circles  on  the 
segments  AiBi  and  CiDi  as  diameters.  These  circles  intersect  in  two 
points  PI  and  Qi  (why?).  The  sheaves  of  rays  whose  centers  are  PI 
and  Qi  correspond  to  sheaves  in  the  other  plane  to  which  they  are 
also  congruent,  each  to  each. 

138.  The  Affinity. — Two  collinearly  related  planes  whose  ideal 
lines  correspond  to  each  other  are  said  to  be  affinately  related  to 
each  other,  and  the  correspondence  existing   between   the   two 
planes  is  called  an  affinity. 

An  affinity  can  be  established  between  two  planes  by  choosing 
the  sides  of  any  triangle  in  one  to  correspond  to  the  sides  of  any 
triangle  in  the  other,  each  to  each.  For  these  triangles,  together 
with  the  ideal  line  in  each  plane,  form  two  complete  quadrilaterals 
whose  sides  correspond,  each  to  each.  A  collineation  is  thus  es- 
tablished between  the  two  planes  (Art.  131)  such  that  the  ideal 
line  in  one  corresponds  to  the  ideal  line  in  the  other. 

139.  Fundamental    Property    of   an   Affinity. — Corresponding 
point-rows  in  affinately  related  planes  are  similarly  projective 
(Art.  61),  that  is,  corresponding  segments  bear  a  constant  ratio 
to  each  other.     The  following  fundamental  property  is  a  conse- 
quence of  this  fact. 

In  affinately  related  planes,  the  ratio  between  the  areas  of  corre- 
sponding figures  is  constant. 

For  a  parallelogram  in  one  plane  must  correspond  to  a  parallelo- 
gram in  the  other,  since  the  ideal  line  in  one  corresponds  to  the 
ideal  line  in  the  other.  Suppose  (Fig.  105)  the  parallelograms 
ABCD  and  EFGH  in  a  correspond  respectively  to  the  parallelo- 
grams, A  iBiCiDi  and  EiFiGiHi  in  a\.  It  follows  that  the  parallelo- 
gram KLMN,  constructed  as  in  the  figure,  corresponds  to  the 
parallelogram  K\LiMiN\.  Since  the  areas  of  parallelograms  hav- 


§139]      PRIMITIVE  FORMS  OF  THE  SECOND  KIND        171 

ing  the  same  altitude  are  to  each  other  as  the  lengths  of  corre- 
sponding sides,  we  have  the  following  proportions: 

ABCD  _  AB  A1B1C,D1  _  AjBj 

KLMN  ~  LK'  KAiMiNi  "  L^' 

EFGH  E  F  E\FiGiHi  E\F\ 

KLMN  =  LM'  K^MiN^  =  L^Mi 


FIG.  105. 


and  hence, 

ABCD _  AB 
EFGH  ~  EF' 

But, 


Ni 


CjDi  _  A1Bl 
iG\H\       E\F\ 


AB  _  AiBt 
EF  ~  EiFi 


and  therefore, 

AE_CJ^  _  EFGH 

A        T">     S~1      T\         TTT     T1^    S*1     TT         *""      •'* 


Since  a  triangle  can  be  regarded  as  half  a  parallelogram,  we  see 
that  the  areas  of  corresponding  triangles  are  to  each  other  in  the 
same  constant  ratio  m. 

Corresponding  polygons  can  be  divided  into  series  of  corre- 
sponding triangles  and  hence,  by  composition  of  ratios,  their  areas 
are  to  each  other  in  the  same  constant  ratio  m. 

Finally,  the  areas  of  corresponding  curves  can  be  regarded  as 
the  limits  of  corresponding  series  of  circumscribed,  or  inscribed, 


172  PROJECTIVE  GEOMETRY  [§140 

polygons  and  hence  bear  the  same  constant  ratio  to  each  other. 
We  conclude,  therefore,  that  if  A  and  AI  represent  the  areas  of 
any  two  corresponding  figures  in  two  affinately  related  planes,  we 
always  have 

A 

A1  =  m- 

140.  Corresponding  Conies  in  Affinately  Related  Planes.— Cor- 
responding conies  in  affinately  related  planes  are  always  of  the 
same  kind,  since  the  ideal  points  on  one  conic  must  correspond  to 
the  ideal  points  on  the  other.  Thus,  ellipses  can  only  correspond 
to  ellipses  (or  to  circles),  parabolas  to  parabolas,  and  hyperbolas 
to  hyperbolas. 

Conversely,  two  conies  of  the  same  kind  can  be  correlated  point 
to  point  in  many  ways  so  that  they  shall  be  corresponding  conies 


in  affinately  related  planes.  Thus,  suppose  k  and  ki  (Fig.  106) 
are  two  parabolas,  correlated  point  to  point,  and  that  A  and  B 
correspond  respectively  to  A\  and  B\.  The  tangents  at 
these  pairs  of  points  intersect  in  the  points  C  and  C\.  The 
triangles  ABC  and  AiBiCi  serve  to  determine  an  affinity  be- 
tween the  planes  of  k  and  A^  (Art.  138)  in  which  to  the  parabola  k 
must  correspond  a  parabola  k'  touching  AiCi  in  AI  and  .Bid  in 
Bi.  But  k'  coincides  with  k  i,  since  it  has  three  tangents  and  the 
points  of  contact  on  two  of  them  in  common  with  ki. 


§142]      PRIMITIVE  FORMS   OF  THE  SECOND  KIND       173 

141.  The  Area  of  a  Parabolic  Segment. — The  area  of  any  para- 
bolic segment  is  two-thirds  the  area  of  the  triangle  whose  sides  are  the 
chord  of  the  segment  and  the  tangents  at  the  extremities  of  the  chord. 

For,  if  (AB)  represents  the  area  of  the  parabolic  segment  whose 
chord  is  AB  (Fig.  106),  we  have  the  following  proportion: 
(AB)     _  ABC 
(A&)  ~  A^d' 

But  we  can  establish  an  affinity  between  the  planes  of  k  and  ki  by 
choosing  any  other  triangle  formed  like  ABC  to  correspond  to 
AiBiCi.     Let  D  be  the  point  where  the  diameter  through  C  meets 
the  parabola,  and  let  FG  be  the  tangent  at  D.     Now  in  the  affinity 
determined  by  the  triangles  ADF  and  AiBiCi  we  have 
(AD)         ADF 
(AiBi)  ~  AiB^' 
Similarly, 

(DB)        DBG 


Hence, 

(AB)  _  (AD)  _  (DB)  _ 
ABC  ~  ADF  ~  DBG  ~ 
The  constant  m  can  be  found  by  means  of  the  equation 

(AB)  =  (AD)  +  (DB)  +  ADB.  (1) 

For,  since  D  bisects  CE  (Art.  75,  5),  ADB  =  %ABC.     Also, 

FCG  =  ADF  +  DBG  =  y±ABC. 
Hence,  from  (1), 

m-ABC  =  m-ADF  +  m-DBG  +  %ABC, 
or 

m-ABC  =  --ABC  +  y2ABC. 

Therefore,  m  =  %  and  (AB)  =  %ABC. 

142.  The  Theorem  of  Apollonius. — When  two  ellipses,  or  two 
hyperbolas,  are  corresponding  conies  in  two  affinately  related 
planes,  any  diameter  of  the  one  must  correspond  to  a  diameter  of 
the  other.  For  a  system  of  parallel  chords  of  the  one  corresponds 
to  a  system  of  parallel  chords  of  the  other.  The  mid-points  of 
the  one  system  must  correspond  to  the  mid-points  of  the  other 
system,  since  the  mid-point  of  any  chord  is  the  harmonic  conjugate 


174 


PROJECTIVE  GEOMETRY 


l§142 


of  the  ideal  point  on  that  chord  with  respect  to  the  curve  points. 
Hence  diameters  correspond  to  diameters.  It  follows  also  that 
a  pair  of  conjugate  diameters  of  the  one  curve  always  corresponds 
to  a  pair  of  conjugate  diameters  of  the  other. 

Suppose  k  and  k\  are  two  ellipses  in  the  same  or  in  different 
planes  (Fig.  107),  and  suppose  AC,  BD  and  ^iCi,  BiDi  are  pairs  of 
conjugate  diameters  of  k  and  kt  respectively.  In  the  affinity  de- 
termined by  the  triangles  ABC  and  AiBiCi,  the  two  ellipses  must 
correspond  point  to  point,  since  the  conic  corresponding  to  k  must 
be  an  ellipse  having  AiCi  and  BiDi  for  a  pair  of  conjugate  diame- 
ters and  it" must  pass  through  AI,  B\,  and  C\  and  hence  coincides 
with  ki. 

Theorem  of  Apollonius. — The  area  of  any  parallelogram  inscribed 
in  an  ellipse  and  whose  diagonals  are  conjugate  diameters  is  lab, 
where  a  and  b  are  the  lengths  of  the  semi-axes. 

B  M 


107. 


For,  suppose  E  and  E\  represent  the  areas  of  two  ellipses  which 
correspond  to  each  other  in  affinately  related  planes. 
Then  (Fig.  107),    ' 

A  B  C  D        E          .  n-.~i       E     .  .   n  „  ^  . 
Aji^Dt  =  El'  or  ABCD  =  E[  •  (A^C^- 
The  right-hand   member  of   this  equation   is   constant   for  all 
parallelograms  formed  like  ABCD.     Therefore,  these  parallelo- 
grams all  have  the  same  area.     But  the  parallelogram  whose 
diagonals  are  the  axes  of  the  ellipse  is  formed  like  ABCD  and  its 
area  is  2ab,  where  a  and  6  are  the  lengths  of  the  semi-axes.     Hence 
the  theorem. 

The  theorem  of  Apollonius  is  at  once  extended  to  parallelo- 
grams circumscribed  about  an  ellipse  and  whose  sides  are  parallel 


§142]      PRIMITIVE  FORMS  OF  THE  SECOND  KIND        175 

to  pairs  of  conjugate  diameters,  since  the  area  of  such  a  parallelo- 
gram is  twice  the  area  of  the  corresponding  inscribed  parallelogram. 
In  the  figure,  the  area  of  KLMN  is  twice  the  area  of  ABCD. 
Hence  : 

The  area  of  any  parallelogram  circumscribed  about  an  ellipse 
and  whose  sides  are  parallel  to  a  pair  of  conjugate  diameters  is 
4a6,  where  a  and  b  are  the  lengths  of  the  semi-axes. 

From  either  of  the  above  theorems,  we  can  at  once  determine 
the  area  of  an  ellipse.  For  we  can  establish  an  affinity  between 
the  plane  of  the  ellipse  and  the  plane  of  ar.y  circle  so  that  the 
two  curves  shall  correspond  point  to  point.  If  E  and  P  represent 
respectively  the  area  of  the  ellipse  and  the  area  of  a  parallelogram 
circumscribed  about  the  ellipse  whose  sides  are  parallel  to  a 
pair  of  conjugate  diameters,  while  E\  and  PI  stand  for  the  area 
of  the  circle  and  the  area  of  the  circumscribed  square  which  corre- 
sponds to  the  parallelogram  P,  then 


E!       Pi 

But  PI  =  4r2(r  being  the  radius  of  the  circle),  P  =  4a6,  and 
EI  =  TiT2.     Substituting  in  the  above  equation,  we  have 

E  =  irab. 

Exercises 

1.  Show  that  the  area  of  any  triangle  inscribed  in  a  parabola  is 
twice  the  area  of  the  triangle  whose  sides  touch,  the  parabola  at  the 
vertices  of  the  inscribed  triangle. 

2.  Two  planes  are  superposed,  determine  the  axis  and  center  of  a 
perspectivity  between  them  which  will  transform  a  given  triangle  in 
one  into  an  equilateral  triangle  in  the  other. 

Suggestions.  —  Let  ABC  be  the  given  triangle  and  M  the  mid-point 
of  the  side  AC.  Draw  the  median  BM  and  parallels  to  it  through 
A  and  C.  Erect  any  perpendicular  to  BM  and  let  it  meet  the  parallels 
in  A  i  and  C\.  Construct  an  equilateral  triangle  on  AiCi  as  one  side 
and  let  Bi  be  the  third  vertex.  The  triangles  ABC  and  AiBiCi  are 
in  perspective  position. 

3.  Show  that  the  perspectivity  determined  in  the  last  exercise  is  an 
affinity. 

4.  Show  that  in  two  superposed  and  perspectively  related  planes 
any  conic  is  transformed  into  itself  provided  it  passes  through  one 


176  PROTECTIVE  GEOMETRY  [§143 

pair  of  corresponding  points  and  the  center  and  axis  of  perspectivity 
are  pole  and  polar  line  with  respect  to  the  conic.  Show  also  that  the 
perspectivity  is  necessarily  an  involution. 

6.  If  two  hyperbolas  correspond  to  each  other  in  affinately  related 
planes,  show  that  the  asymptotes  of  one  correspond  to  the  asymptotes 
of  the  other. 

6.  If  k  and  k\  are  two  hyperbolas  in  the  same  plane,  or  in  different 
planes,  and  we  choose  the  triangle  formed  by  the  asymptotes  and  any 
third  tangent  of  the  one  to  correspond  to  the  triangle  formed  by  the 
asymptotes  and  any  third  tangent  of  the  other  each  to  each,  show 
that  an  affinity  is  determined  between  the  planes  in  which  the  hyper- 
bolas correspond  point  to  point. 

143.  The  Similitude. — If  in  two  affinately  related  planes,  the 
corresponding  ideal  point-rows  are  congruent;  that  is,  if  they  can 
be  superposed  so  as  to  coincide  point  to  point,  the  planes  are  said 
to  be  similar,  and  the  collineation  existing  between  them  is  called  a 
similitude. 

When  two  similar  planes  are  superposed,  the  similitude  ex- 
isting between  them  is  called  direct  or  inverse  according  as  their 
ideal  point-rows  are  directly  or  oppositely  projective  (Art.  40). 

When  two  similar  planes  are  superposed  and  in  perspective 
position;  that  is,  when  their  ideal  point-rows  coincide  point  to 
point,  the  center  of  perspectivity  is  called  the  center  of  simili- 
tude, and  the  collineation  is  called  a  perspective  similitude. 

144.  Properties  of  the  Similitude. — 

1.  In  a  similitude,  corresponding  angles  are  equal. 

For,  if  P  corresponds  to  PI,  and  the  ideal  points  A  and  B  corre- 
spond respectively  to  the  ideal  points  AI  and  B\,  the  angle  APB 
is  equal  to  the  angle  AiPiBi,  since  the  planes  can  be  superposed  so 
that  the  points  A  and  B  coincide  respectively  with  AI  and  B\. 

2.  In  a  similitude,  corresponding  figures  are   similar  figures. 
For,  if  ABC  and  AiBiCi  are  corresponding  triangles,  they  are 

equiangular  by  property  1,  and  consequently  similar. 

Two  corresponding  polygons  can  be  divided  into  sets  of  corre- 
sponding triangles  and  hence  are  similar  polygons. 

Corresponding  curves  can  be  considered  as  the  limits  of  sets 
of  corresponding  inscribed,  or  circumscribed,  polygons  and 
are  consequently  similar  curves.  In  particular,  the  circles 
in  one  plane  correspond  to  circles  in  the  other. 


§145]      PRIMITIVE  FORMS  OF  THE  SECOND  KIND        177 

3.  If  a  collineation  between  two  planes  transforms  the  circles 
in  one  plane  into  circles  in  the  other,  the  collineation  is  a  similitude. 

For  the  ideal  line  in  one  plane  must  correspond  to  the  ideal  line 
in  the  other,  otherwise  the  circles  in  one  plane  which  cut  the  limit- 
ing line  in  that  plane  would  be  transformed  into  hyperbolas  in  the 
other  plane,  contrary  to  hypothesis.  The  collineation  is,  therefore, 
an  affinity,  and  consequently  to  each  pair  of  perpendicular 
(conjugate)  diameters  of  any  circle  must  correspond  a  pair  of  per- 
pendicular diameters  of  the  corresponding  circle  (Art.  142). 
Hence  the  collineation  is  a  similitude. 

4.  A  similitude  is  determined  between  two  planes  by  choosing  any 
triangle  in  one  to  correspond  to  a  similar  triangle  in  the  other. 

For,  in  the  affinity  determined  by  these  two  triangles,  the  cor- 
responding ideal  point-rows  are  congruent. 

5.  Any  two  circles  in  a  plane  correspond  to  each  other  in  two 
definite  perspective  similitudes.     One  of  these  is  direct  and  corre- 
sponding parallel  radii  are  drawn  in  the  same  direction;  the  other  is 
inverse  and  corresponding  parallel  radii  are  drawn  in  opposite 
directions. 

Let  0  and  0i  be  the  centers  of  any  two  circles  k  and  ki,  respec- 
tively (Fig.  108).  Draw  any  pair  of  parallel  diameters  as  AC  and 
AiCi  and  let  AAi  and  OOi  meet  in  S.  The  similar  triangles  AOS 
and  AiOiS  determine  a  direct  perspective  similitude  which  trans- 
forms k  into  ki.  If  AC i  meets  00 'i  in  S',  then  the  triangles  AOS' 
and  C\OiS'  determine  an  inverse  perspective  similitude  which 
likewise  transforms  k  into  k\. 

S  is  called  the  direct  center  of  similitude  and  Sr,  the  inverse 
center  of  similitude. 

145.  Inverse  Points.  Radical  Axis. — Any  secant  through  a 
center  of  similitude  S  (Fig.  108)  cuts  a  pair  of  corresponding  circles 
k  and  ki  in  two  pairs  of  corresponding  points  A,  AI  and  B,  B\. 
Two  points  cut  by  the  same  secant,  one  from  each  circle,  but  which 
do  not  correspond  to  each  other  are  called  inverse  points.  Thus 
B  and  AI  are  inverse  points.  So  also  are  A  and  BI. 

If  the  points  on  k  are  correlated  to  their  respective  inverse 

points  on  ki  a  perspectivity  is  determined  in  the  plane  whose 

center  coincides  with  S  and  which  transforms  k  into  ki.     The  axis 

of  this  perspectivity  is  called  the  radical  axis  of  the  two  circles. 

12 


178 


PROJECTIVE  GEOMETRY 


[§146 


Tangents  at  corresponding  inverse  points  intersect  upon  the 
radical  axis.  Thus  the  radical  axis  of  the  two  circles  k  and  ki  is  the 
line  TQ. 

The  segments  of  the  tangents  to  two  circles  contained  between  any 
point  of  the  radical  axis  and  the  points  of  contact  are  equal  in  length. 


FIG.  108. 

Let  any  secant  through  S  meet  the  radical  axis  in  the  point  U 
and  the  points  of  contact  of  tangents  from  U  be  M  and  N.  The 
tangents  at  A  and  AI  are  parallel  to  each  other.  So  also  are  the 
tangents  at  B  and  BI.  Hence,  the  triangles  TUAi  and  QUA  are 
similar.  Likewise,  the  triangles  TUB  and  QUB\  are  similar. 
Therefore, 

UAi  _  UA         UBi      UB, 

TAi  ~  AQ         QBi  ~  BT 

But  TBAi  and  AQBi  are  isosceles  triangles,  since  each  has  a  pair 
of  equal  angles,  and  hence  TA\  =  BT  and  AQ  =  QBi. 
It  follows  that, 

UA1UB1  =  UA-UB, 
and  therefore, 

UN  =  UM. 

146.  The  Congruence. — A  special  similitude  is  the  congruence 
in  which  corresponding  similar  triangles  are  congruent  triangles. 
Two  congruent  planes  can  be  superposed  so  as  to  coincide  through- 
out. A  congruence  is  either  direct  or  inverse. 


§147]      PRIMITIVE  FORMS  OF  THE  SECOND  KIND        179 

Just  as  the  affinity  is  a  special  collineation  and  the  similitude  is 
a  special  affinity,  so  is  the  congruence  a  special  similitude. 

Exercises 

1.  Show  that  a  circle  can  be  drawn  to  touch  two  given  circles  in  a 
pair  of  inverse  points. 

2.  If  a  circle  is  drawn  to  touch  two  given  circles,  show  that  the  line 
joining  the  points  of  contact  will  pass  through  a  center  of  similitude. 

3.  Three  circles,  taken  two  and  two,  determine  three  radical  axes. 
Show  that  these  radical  axes  meet  in  a  point.     This  point  is  called  the 
radical  center  of  the  three  circles. 

4.  Two  given  circles  are  external  to  each  other  and  cut  the  line  join- 
ing their  centers  in  the  points  A,  B  and  C,  D.     Any  circle  whose  cen- 
ter is  a  point  0  on  the  radical  axis  of  the  two  given  circles  and  whose 
radius  is  equal  to  the  length  of  the  tangent  from  O  to  either  of  the  given 
circles  will  cut  the  given  circles  orthogonally  and  the  line  ABCD  in 
two  points  which  separate  harmonically  both  A,  B  and  C,  D. 

6.  The  radical  axis  of  any  two  circles,  each  of  which  touches  two 
given  circles,  passes  through  a  center  of  similitude  of  the  given  circles. 

6.  One  circle  is  inside  another  but  not  concentric  with  it.     Con- 
struct the  centers  of  similitude. 

7.  Two  circles  intersect.     Construct  the  centers  of  similitude  and 
show  that  the  radical  axis  is  the  common  chord. 

8.  Show  that  the  perspectivity  that  transforms  the  points  on  one 
circle  into  the  inverse  points  on  another  is  composed  of  a  similitude 
which  transforms  the  first  circle  into  the  second  and  an  involution 
which  leaves  the  second  circle  unchanged. 

9.  Show  that  the  invariant  (Art.  135)  of  a  perspective  similitude  is 
positive  or  negative  according  as  the  similitude  is  direct  or  inverse. 

10.  The  invariant  of  a  perspective  congruence  is  +  1  or  —  1  ac- 
cording as  the  congruence  is  direct  or  inverse. 

11.  Show  that  the  resultant  of  two  perspectivities  in  the  same 
plane  which  have  a  common  center  is  a  third  perspectivity  having  the 
same  center  and  whose  axis  is  concurrent  with  the  axes  of  the  origi- 
nal perspectivities.     Show  also  that  the  invariant  of  the  resultant  per- 
spectivity is  the  product  of  the  invariants  of  the  original  perspectivities. 

147.  Collineation  in  the  Plane.  Self-corresponding  Ele- 
ments.— 

1.  When  two  collinearly  related  planes  are  superposed  but  are  not 
in  perspective  position,  they  cannot  have  more  than  three  self-cor- 
responding points  unless  they  coincide  element  to  element. 


180 


PROJECTIVE  GEOMETRY 


[§147 


For  suppose  that  four  points  A,  B,  C,  D  coincide  with  their 
corresponding  points  Ai,  Bi,  Ci,  Z>i.  No  three  of  these  points  can 
lie  upon  the  same  straight  line,  for  then  this  line  would  be  a  self- 
corresponding  point-row  and  the  planes  in  perspective  position,  con- 
trary to  hypothesis.  Hence  the  points  A,B,C,D  are  the  vertices 
of  a  complete  quadrangle.  The  sheaf  whose  center  is  A  coincides 
with  the  sheaf  whose  center  is  Ai,  since  the  rays  AB,  A  C,  AD  coin- 
cide with  their  corresponding  rays.  Similarly,  the  sheaves  B  and 
BI  coincide.  Any  point  P  of  the  plane  must  coincide  with  its 
corresponding  point  PI,  since  the  rays  joining  P  to  A  and  B  coin- 
cide with  their  corresponding 
rays.  Therefore,  the  planes 
coincide  element  to  element. 

On  the  other  hand,  it  is  evi- 
dent that  a  collineation  can 
have  three  self-corresponding 
points.  For,  if  we  choose  the 
vertices  of  a  complete  quad- 
rangle A,  B,  C,  D  to  correspond 
respectively  to  the  vertices  A, 
B,  C,  DI  of  another  complete 
quadrangle,  a  collineation  is 
completely  and  uniquely  de- 
termined (Art.  131)  which  has  the  three  points  A,  B,  C  as  self- 
corresponding  points. 

If  a  collineation  in  a  plane  has  three  self-corresponding  points 
forming  a  triangle  ABC,  the  sides  of  this  triangle  are  self-corre- 
sponding lines,  but  the  point-rows  along  these  lines  do  not  coincide 
point  to  point.  The  point-rows  along  AB,  for  example,  are  pro- 
jectively  related  and  have  A  and  B  as  self-corresponding  points. 
2.  Two  superposed  and  collinearly  related  planes,  not  in  per- 
spective position  nor  coincident,  must  have  at  least  one  self-corre- 
sponding point  and  one  self-corresponding  line. 

Since  the  collineation  is  not  a  perspectivity,  there  are  points  in  the 
plane  which  do  not  correspond  to  themselves  nor  lie  on  any  sself- 
corresponding  line.  Let  A  be  such  a  point  (Fig.  109).  Suppose 
A  corresponds  to  AI  and,  in  turn,  AI  corresponds  to  A 2.  The 
points  A,  A i,  A 2  cannot  lie  on  the  same  line,  for  then  this  line  would 
be  a  self-corresponding  line. 


FIG.  109. 


§147]      PRIMITIVE  FORMS  OF  THE  SECOND  KIND       181 

The  protectively  related  sheaves  whose  centers  are  A  and  AI 
generate  a  conic  k,  and  the  sheaves  whose  centers  are  AI  and  A2,  a 
second  conic  ki.  The  conic  k  passes  through  A  and  A  i  and  touches 
AiA2  at  AI  since  the  line  A\A2  corresponds  to  AAi.  Likewise, 
the  conic  kt  passes  through  AI  and  A2  and  touches  AAi  at  AI. 
The  conies  do  not  touch  each  other  at  AI,  since  their  tangents  at 
this  point  do  not  coincide.  Hence,  the  conic  k  crosses  ki  at  AI. 
The  point  A  is  necessarily  outside  ki,  and  therefore  k  is  partly  out- 
side and  partly  inside  ki,  and  must  consequently  cross  ki  in  at  least 
one  other  point  besides  AI.  This  second  point  P,  whose  existence 
is  thus  demonstrated,  is  a  self-corresponding  point,  since  the  ray 
AP  corresponds  to  the  ray  A\P,  and,  in  turn,  the  ray  AiP  corre- 
sponds to  the  ray  AjP. 

The  two  conies  k  and  ki  serve  to  construct  the  point  correspond- 
ing to  any  given  point  in  the  plane.  Thus,  if  X  is  any  point  in  the 
plane,  the  line  AX  corresponds  to  the  line  AiF,  the  two  lines 
meeting  in  the  point  M  of  the  conic  k.  Again,  the  line  Ai-X"  cor- 
responds to  the  line  A2Y,  the  two  lines  meeting  in  the  point  N  of 
the  conic  ki.  The  point  Y,  corresponding  to  X,  is  the  intersection 
of  the  lines  A\M  and  A2N. 

By  means  of  the  principle  of  duality,  each  step  of  the  foregoing 
proof  can  be  replaced  by  its  reciprocal,  and  we  thus  see  that  a 
collineation  in  a  plane  must  have  at  least  one  self-corresponding 
line. 

The  conies  k  and  ki  can  meet  in  at  most  four  points  without 
coinciding.  If  these  points  are  AI,  P,  Q,  R,  then  P,  Q,  R  are  self- 
corresponding  points  of  the  collineation,  and  the  sides  of  the 
triangle  PQR  are  self-corresponding  lines. 

Two  of  the  points  P,  Q,  R  may  coincide  in  a  single  point.  The 
conies  k  and  ki  then  touch  each  other  at  this  point,  and  the  col- 
lineation has  two  self-corresponding  points  and  two  self-corre- 
sponding lines.  One  of  these  lines  joins  the  self -corresponding 
points,  and  the  other  passes  through  one  of  them  and  touches  both 
conies  there. 

The  results  of  the  foregoing  discussion  are  brought  together  in 
the  following  theorem. 

3.  Two  superposed  and  collinearly  related  planes,  not  in  perspec- 
tive position  nor  coincident,  have  either  one  point  and  one  line  as  self- 


182  PROJECTIVE  GEOMETRY  [§147 

corresponding  elements;  or  two  points  and  two  lines  as  self-corre- 
sponding elements,  in  which  case  one  self-corresponding  line  joins  the 
self-corresponding  points  and  the  other  passes  through  one  of  them; 
or  the  vertices  and  sides  of  a  triangle  as  self-corresponding  elements. 

Exercises 

1.  Given  two  conies  k  and  ki  in  the  same  plane.     Construct  the 

polar  lines  a,  b,  c, of  a  series  of  points  A,  B,  C, with  respect  to 

k  and  the  poles  A\,  BI,  Ci, uf  a,  b,  c, with  respect  to  ki.     Show 

that  the  correspondence  between  the  points  A,  B,  C, and  Ai,  BI, 

Ci, is  a  collineation  having  as  self -corresponding  elements  the  ver- 
tices and  sides  of  the  self-polar  triangle  common  to  k  and  ki. 

2.  Draw  two  conies  intersecting  in  four  points.     Choose  one  of 
these  points  for  the  point  A\  in  Fig.  109  and  construct  a  number  of 
corresponding  elements  of  the  collineation  determined  by  the  conies. 
In  particular,  show  that  the  remaining  three  points  of  intersection  of 
the  two  conies  are  self-corresponding  points  of  the  collineation. 

3.  Prove  that  a  collineation  in  a  plane  must  have  at  least  one  self- 
corresponding  line  by  means  of  the  suggestion  in  Art.  147. 

4.  If  in  a  plane,  the  vertices  A,  B,  C,  D  of  a  complete  quadrangle 
correspond  respectively  to  the  points  A,  B,  C,  D\,  show  how  to  con- 
struct the  point  PI  corresponding  to  any  point  P  in  the  plane. 


CHAPTER    XV 

POLARITIES  IN  A  PLANE  AND  IN  A  BUNDLE 

148.  The  Polarity  in  a  Plane. — When  two  planes  are  reciprocally 
related  and  superposed  they  form  a  duality  in  their  common 
plane  (Art.  128).  This  duality  does  not,  in  general,  correlate  the 
elements  of  the  plane  so  that  any  two  corresponding  elements, 
point  and  line,  correspond  to  each  other  doubly.  That  is,  if  the 
superposed  planes  are  a  and  «i,  and  the  line  a  of  a  corresponds 
to  the  point  AI  of  a\,  it  is  not  true,  in  general,  that  A\  of  a 
corresponds  to  a  of  a\.  If,  however,  the  duality  is  such  that  every 
pair  of  corresponding  elements  is  doubly  corresponding,  the  duality 
is  involutoric,  and  is  called  a  polarity. 

That  polarities  exist  in  a  plane  is  evident,  since  the  corre- 
spondence between  pole  and  polar  line  with  respect  to  a  fixed 
conic  is  involutoric  and  forms  a  duality  in  the  plane. 

Two  elements,  point  and  line,  that  correspond  to  each  other 
in  a  polarity  are  called  pole  and  polar  line  respectively.  Since 
the  superposed  planes  constituting  the  polarity  are  projectively 
related,  a  pair  of  incident  elements  in  one  corresponds  to  a  pair 
of  incident  elements  in  the  other.  Hence: 

Of  two  points  in  a  polarity,  either  each  or  neither  lies  upon  the 
polar  line  of  the  other;  and  of  two  lines,  either  each  or  neither  passes 
through  the  pole  of  the  other. 

Two  points  in  a  polarity  are  conjugate  points  if  each  lies  on  the 
polar  line  of  the  other,  and  two  lines  are  conjugate  lines  if  each 
passes  through  the  pole  of  the  other. 

A  point  is  self-conjugate  if  it  lies  on  its  own  polar  line,  and  a 
line  is  self -conjugate  if  it  passes  through  its  own  pole. 

To  every  figure  in  the  plane  corresponds  a  polar-figure.  To 
a  triangle  corresponds  a  polar  triangle ;  to  a  curve  of  second  order, 
an  envelope  of  second  class;  and  so  on. 

A  triangle  is  self -polar  if  each  side  is  the  polar  line  of  the 
opposite  vertex. 

183 


184  PROJECTIVE  GEOMETRY  [§149 

149.  Construction  of  a  Polarity  in  a  Plane. — 

A  'polarity  is  determined  having  given  a  self-polar  triangle  and  a 
point,  not  on  any  of  the  sides  of  the  triangle,  together  with  its  corre- 
sponding line,  not  passing  through  any  of  the  vertices  of  the  triangle. 

It  is  evident  that  the  given  elements  determine  a  duality 
in  the  plane.  For,  if  A,  B,  C,  and  a,  b,  c  are  respectively  the  ver- 
tices and  opposite  sides  of  the  given  triangle,  and  D  and  d  are 
the  given  point  and  its  corresponding  line  (Fig.  110),  then  A, 
B,  C,  D  form  the  vertices  of  a  complete  quadrangle  and  a,  b,  c,  d, 


FIG.  110. 

the  sides  of  the  corresponding  complete  quadrilateral  (Art.  131). 
We  have  now  to  show  that  this  duality  is  involutoric  and  hence 
forms  a  polarity.  Join  D  to  the  vertices  A,  B,  C  by  the  lines 
k,  m,  n  respectively.  To  these  lines  correspond  the  points  K, 
M,  N  where  d  meets  the  sides  a,  b,  c.  Let  the  lines  k,  m,  n  meet 
the  sides  a,  b,  c  in  the  points  S,  T,  R.  To  these  points  correspond 
the  lines  s,  t,  r  joining  A,  B,  C  to  K,  M,  N  respectively.  From 
the  definition  of  projectivity  we  have, 

SCKB X  sckb A  KBSC— 

Therefore, 

SCKB A  KBSC 

and  we  see  that  along  the  side  a  of  the  self-polar  triangle  there 
is  an  involution  in  which  any  point  corresponds  to  the  intersection 
of  its  corresponding  line  with  a.  Similarly,  along  the  sides  b 
and  c  exist  involutions  formed  in  like  manner. 


§150]    POLARITIES  IN  A  PLANE  AND  IN  A  BUNDLE     185 

If  p  is  any  line  in  the  plane  meeting  a,  b,  c  in  the  points  K' , 
M',  Nf,  then  these  points  correspond,  in  the  involutions  on  a,  b, 
c,  to  three  points  S',  T',  R',  such  that  AS',  BT',  CR'  meet  in  the 
point  P  corresponding  to  the  line  p.  But  the  line  corresponding 
to  P  must  be  p,  since  the  lines  AP,  BP,  CP  meet  the  sides  a,  b,  c 
in  the  points  S',  T',  R'  and  these  correspond,  in  the  involutions 
on  a,  b,  c,  to  K',  M',  N'  respectively.  Hence  P  and  p  correspond 
doubly  and  the  duality  is,  therefore,  a  polarity. 

150.  Self-conjugate  Points  in  a  Polarity. — If  the  involution 
on  a  side  of  a  self-polar  triangle  is  hyperbolic,  the  foci  of  the  in- 


FIG.  111. 

volution  are  self-conjugate  points  in  the  polarity,  since  a  focus 
lies  on  its  polar  line. 

1.  There  is  no  line  of  a  polarity  all  of  whose  points  are  self-con- 
jugate; and  no  point  all  of  whose  rays  are  self-conjugate. 

For,  suppose  p  is  such  a  line  (Fig.  111).  Let  Q  and  R  be 
any  two  points  on  it,  and  let  P  be  its  pole.  The  polar  lines 
of  Q  and  R  are  the  lines  PQ  and  PR  respectively.  Any  point  on 
PQ,  as  S,  has  for  its  "polar  line  a  line  s  passing  through  Q  and  meeting 
PR  in  T.  The  polar  line  of  T  joins  S  and  R,  meeting  QT  in  L. 
The  polar  line  of  L  is  ST,  meeting  p  in  M.  Now  the  polar  line 
of  M  is  PL  which  cannot  pass  through  M,  since  M  is  the  intersec- 


186  PROJECTIVE  GEOMETRY  [§150 

tion  of  two  diagonals  of  the  complete  quadrangle  LSPT  and  PL 
is  the  third  diagonal.  Hence,  M  is  not  a  self-conjugate  point, 
contrary  to  the  supposition  made.  We  conclude,  therefore,  that 
no  line  can  have  all  its  points  self-conjugate. 

The  principle  of  duality  enables  us  to  say  that,  reciprocally, 
there  is  no  point  in  the  plane  all  of  whose  rays  are  self-conjugate 
lines. 

2.  In  a  given  polarity  in  a  plane  there  are  an  infinity  of  self- 
polar  triangles. 

For,  let  p  be  any  line  in  the  plane  and  P  its  pole  (Fig.  112). 

Upon  p  choose  any  point  Q 
which  is  not  a  self-conjugate 
point  (cf.  1).  The  polar  line 
of  Q  passes  through  P  and 
meets  p  in  a  point  R  distinct 
from  Q.  The  polar  line  of  R  is 
then  PQ,  and  PQR  is  a  self-polar 

FIG.  112.  triangle.      Hence     a     self -polar 

triangle     can     always    be    con- 
structed having  any  line  of  the  polarity  as  one  side.     Indeed  an 
infinity  of  such  triangles  can  be  constructed  having  p  as  one  side, 
since  Q  can  be  any  point  on  p  which.is  not  self-conjugate. 
It  follows  from  this  proposition  and  Art.  149  that: 

3.  Along  any  line  p  of  a  polarity  in  a  plane  there  exists  an  in- 
volution in  which  any  point  corresponds  to  the  intersection  of  its 
polar  line  with  p;  that  is,  to  its  conjugate  point  on  p. 

If  this  involution  is  hyperbolic,  its  foci  are  self-conjugate  points 
of  the  polarity.  Hence : 

4.  Not  more  than  two  self-conjugate  points  of  a  polarity  can  lie 
on  any  line  in  the  plane. 

If  a  line  p  is  self-conjugate,  its  pole  P  is  the  only  self-conjugate 
point  on  it,  since  the  polar  lines  of  all  the  other  points  of  it  must 
pass  through  P.  The  involution  on  a  self-conjugate  line  is  there- 
fore parabolic  or  trivial  (Art.  106). 

5.  Every  self-conjugate  point  S  of  a  polarity  is  one  focus  of  the 
involutions  existing  on  the  lines  passing  through  S. 

6.  The  locus  of  all  the  self-conjugate  points  of  a  polarity  is  a  conic. 
For  no  line  in  the  plane  can  meet  this  locus  in  more  than  two 


§151]    POLARITIES  IN  A  PLANE  AND  IN  A  BUNDLE     187 

points.  These  points  are  the  foci  of  the  involution  on  the  line  and 
are  separated  harmonically  by  any  pair  of  points  in  the  involution 
(Art.  104,  2).  Thus,  in  Fig.  Ill,  V  is  a  self-conjugate  point  of  the 
polarity  on  the  line  s,  since  it  is  separated  harmonically  fr6m  Q  by 
the  conjugate  points  L  and  T.  Similarly,  U  is  a  self-conjugate 
point  on  t.  If  x  is  the  line  RV,  the  conic  generated  by  the  sheaves 


qsp — 

passes  through  V  and  U  and  is  tangent  to  q  and  r  at  the  points  Q 
and  R.  This  conic  is  the  locus  of  self-conjugate  points  of  the 
polarity.  The  triangle  LST  is  self-polar  with  respect  to  this 
conic,  and  the  entire  polarity  appears  as  the  correspondence 
between  pole  and  polar  line  with  respect  to  this  conic. 

The  locus  of  the  self-conjugate  points  of  a  polarity  can  never 
consist  of  two  straight  lines  by  virtue  of  proposition  1. 

The  tangents  to  the  locus  of  self-conjugate  points  are  self- 
conjugate  lines  of  the  polarity,  since  each  tangent  contains  but  one 
self-conjugate  point. 

161.  Classification  of  Polarities  in  a  Plane. — 

In  a  given  polarity  in  a  plane,  the  involutions  on  the  sides  of  any 
self-polar  triangle  are  either  all  elliptic  or  else  one  of  them  is 
elliptic  and  the  other  two  are  hyperbolic. 


K 


FIG.  113. 

Suppose  the  polarity  is  determined  by  the  self-polar  triangle 
ABC  and  a  pair  of  corresponding  elements  p,  P  (Fig.  113).  The 
triangle  ABC  divides  the  plane  into  four  regions  each  of  which  is 
bounded  by  three  segments.  For  example,  the  region  2  is  bounded 
by  the  finite  segment  BC,  the  infinite  segment  CA,  and  the  in- 
finite segment  AB.  The  point  P  must  occupy  one  of  these 
regions  which  we  will  call  the  region  P.  The  polar  line  p  cannot 


188  PROJECTIVE  GEOMETRY  [§152 

pass  through  any  vertex  nor  meet  all  three  of  the  segments  bound- 
ing the  region  P.  If  p  meets  one  of  these  segments  it  must  also 
meet  another,  since  if  it  enters  the  region  P  it  must  pass  out  of  it. 
There  are  then  two  cases  to  consider. 

First,  p  does  not  meet  any  of  the  segments  bounding  the  region 
P.  This  is  the  case  shown  in  the  figure. 

Second,  p  meets  two  of  the  segments  bounding  the  region  P. 

If  p  meets  the  sides  of  the  triangles  a,  6,  c  in  the  points  K,  M,  N, 
respectively,  the  polar  lines  of  these  points  are  the  lines  AP,  BP, 
CP  meeting  the  sides  a,  6,  c  in  the  points  S,  T,  R,  conjugate 
respectively  to  K,  M,  N.  The  points  S,  T,  R  are  necessarily  on 
the  segments  bounding  the  region  P. 

In  the  first  case,  the  points  K,  M,  N  are  none  of  them  on  the 
segments  bounding  the  region  P,  and  hence  the  involution  on  any 
side  of  the  triangle  is  determined  by  two  pairs  of  points  which 
separate  each  other.  For  example,  the  involution  on  the  side  a 
is  determined  by  the  pairs  B,  C  and  K,  S  which  separate  each 
other.  Hence  all  three  of  the  involutions  are  elliptic.  The  in- 
volution along  any  line  of  the  plane  is  necessarily  elliptic,  since  no 
line  can  enter  the  region  containing  its  pole.  There  are,  therefore, 
no  self-con  jugate  points  or  self-conjugate  lines  in  the  plane.  A 
polarity  of  this  type  is  called  uniform. 

In  the  second  case,  two  of  the  three  points  K,  M,  N  are  on  seg- 
ments bounding  the  region  P  while  the  third  is  not.  Hence  on  two 
of  the  sides  a,  b,  c,  the  involutions  are  determined  by  pairs  of  points 
which  do  not  separate  each  other  while  on  the  third  side,  the  in- 
volution is  determined  by  pairs  which  do  separate  each  other. 
Consequently,  in  this  case,  two  involutions  are  hyperbolic  while 
the  third  is  elliptic.  There  are  self-conjugate  points  in  the  plane 
whose  locus  is  a  conic.  Every  self-polar  triangle  in  the  polarity  is 
self -polar  with  respect  to  the  locus  of  self -conjugate  points,  and 
consequently  two  of  the  involutions  on  its  sides  are  hyperbolic  and 
one  is  elliptic.  A  polarity  of  this  type  is  called  non-uniform. 

A  non-uniform  polarity  defines  the  conic  which  is  the  locus  of  its 
self-conjugate  points,  since  this  conic  is  uniquely  determined  by  the 
polarity.  Any  line  in  the  plane  meets  this  conic  in  two  real  points, 
in  one  real  point,  or  in  conjugate  imaginary  points  according  as  the 
involution  along  the  line  is  hyperbolic,  parabolic,  or  elliptic. 


§152]    POLARITIES  IN  A  PLANE  AND  IN  A  BUNDLE     189 

A  uniform  polarity  in  a  plane  defines  an  imaginary  conic  in  the 
plane.  Every  line  of  the  plane  meets  this  conic  in  two  imaginary 
points  defined  by  the  elliptic  involutions  along  the  line. 


Exercises 

1.  Show  that  in  a  polarity  there  is  no  sheaf  of  lines  all  of  whose 
rays  are  self -conjugate  (cf.  Art.  150,  1). 

2.  Show  that  about  any  point  P  of  a  polarity  there  is  an  involution 
in  which  any  ray  corresponds  to  its  conjugate  ray  through  P. 

3.  If  P  is  a  self-conjugate  point  of  a  polarity,  its  polar  line  is  the  only 
self-conjugate  line  through  it. 

4.  In  a  uniform  polarity,  the  involutions  about  the  vertices  of  any 
self -polar  triangle  are  all  elliptic;  in  a  non-uniform  polarity,  two  of 
these  involutions  are  hyperbolic  and  the  third  is  elliptic. 

6.  Show  that  a  polarity  is  determined  having  given  a  self-polar 
triangle,  the  involution  on  one  of  its  sides,  and  a  pair  of  conjugate 
points  not  lying  on  the  sides  of  the  triangle. 

6.  Show  that  a  polarity  is  determined  by  choosing  the  vertices  of 
any  simple  pentagon  to  correspond  respectively  to  the  opposite  sides. 

7.  Show  that  a  polarity  is  determined  having  given  the  involutions 
along  two  non-conjugate  lines  a,  b  and  the  pole  of  one  of  these  lines. 
This  pole  must  lie  on  the  line  joining  the  points  corresponding  to  the 
point  (a  6)  in  the  given  involutions. 

8.  Show  that  a  polarity  is  determined  having  given  the  involutions 
along  two  conjugate  lines. 

9.  If  two  pairs  of  opposite  sides  of  a  complete  quadrangle  are  pairs 
of  conjugate  lines  in  a  polarity,  show  that  the  third  pair  is  a  pair  of 
conjugate  lines  in  the  same  polarity.     (Theorem  of  Von  Staudt-Hesse.) 

162.  The  Polarity  in  a  Bundle. — If  we  project  a  polarity  in  a 
plane  from  a  point  not  lying  in  the  plane,  we  obtain  a  polarity  in 
the  bundle  whose  center  is  the  point  from  which  the  projection  is 
made.  This  polarity  is  uniform  or  non-uniform  according  as  it 
is  the  projector  of  a  uniform  or  a  non-uniform  polarity. 

A  polarity  in  a  bundle  is  an  involutoric  duality  between  the 
lines  and  planes  of  the  bundle.  The  plane  corresponding  to  any 
line  is  the  polar  plane  of  the  line;  and  the  line  corresponding  to 
any  plane  is  the  pole-ray  of  the  plane. 


190  PROJECTIVE  GEOMETRY  [§153 

Two  lines,  each  lying  in  the  polar  plane  of  the  other,  are  conju- 
gate lines ;  two  planes,  each  containing  the  pole-ray  of  the  other, 
are  conjugate  planes. 

A  line  lying  in  its  own  polar  plane  is  self -conjugate ;  a  plane 
passing  through  its  own  pole-ray  is  self -con  jugate. 

All  self-conjugate  lines  of  a  non-uniform  polarity  in  a  bundle 
lie  upon  a  cone  of  second  order,  and  all  self-conjugate  planes  are 
tangent  planes  to  this  cone. 

A  uniform  polarity  in  a  bundle  has  no  real  self-conjugate  lines 
or  self-conjugate  planes.  It  defines  an  imaginary  cone  in  space. 

153.  The  Orthogonal  Polarity. — If  two  orthogonally  correlated 
bundles  (Art.  125)  are  superposed,  that  is,  if  they  are  concentric, 
the  resulting  duality  in  their  common  bundle  is  a  polarity.     For, 
if  a  is  any  line  of  the  bundle,  its  corresponding  plane  a  is  perpen- 
dicular to  it.     The  line  corresponding,  in  turn,  to  a  is  perpendicu- 
lar to  a  and  is,  therefore,  the  line  a.     Any  two  lines  of  the  bundle, 
as  a  and  b,  determine  a  plane  7.     The  polar  planes  of  a  and  b  are 
perpendicular  to  a  and  b  respectively,  and  intersect  in  a  line  c 
which  is  the  pole-ray  of  7  and  is  also  perpendicular  to  7.     Hence 
7  is  the  polar  plane  of  c.     The  duality  in  the  bundle  is,  therefore, 
involutoric  and  thus  forms  a  polarity.     This  polarity  is  called  the 
orthogonal  polarity  in  the  bundle. 

An  orthogonal  polarity  is  necessarily  uniform,  since  no  line  ever 
lies  in  its  polar  plane. 

In  each  plane  of  the  bundle  pairs  of  conjugate  lines  are  so  situ- 
ated that  each  line  of  the  pair  is  perpendicular  to  the  other.  The 
involution  of  con  jugate  lines  in  any  plane  is,  therefore,  circular  (Art. 
108).  Also,  about  any  line  of  the  bundle  pairs  of  conjugate  planes 
are  so  situated  that  each  plane  of  the  pair  is  perpendicular  to  the 
other.  Thus,  the  involution  of  conjugate  planes  about  any  line 
of  the  bundle  may  be  called  a  circular  involution. 

Any  tri-rectangular  pyramid,  whose  vertex  is  the  center  of  the 
bundle,  is  self -polar,  since  the  edges  are  the  pole-rays  of  the  oppo- 
site faces.  Conversely,  any  self-polar  pyramid  is  tri-rectangular. 

154.  Polarity  and  Anti-polarity  with  Respect  to  a  Circle. — A 
given  circle  establishes  a  non-uniform  polarity  in  its  plane  in 
which  two  corresponding  elements,  point  and  line,  are  pole  and 
polar  line  with  respect  to  the  circle.    Let  C  be  the  center  of  the 


§154]    POLARITIES  IN  A  PLANE  AND  IN  A  BUNDLE     191 

circle  and  r,  the  length  of  the  radius  (Fig.  114).  If  A  is  any  point 
in  the  plan,  its  polar  line  a'  meets  the  diameter  through  A  in  a 
point  M'  such  that 

AC  _      r  _    r^ 

r     "  CM7'  01  "  AC' 

If  we  cut  an  orthogonal  polarity  by  any  plane  not  passing 
through  its  vertex  0,  we  obtain  a  uniform  polarity  in  the  plane. 


FIG.  114. 

Let  OC  be  the  perpendicular  from  the  center  of  the  bundle  upon 
the  cutting  plane.  With  OC  as  radius  and  C  as  center,  describe 
a  circle  in  the  plane.  The  lines  of  the  bundle  drawn  to  points 
on  this  circle  are  rays  of  a  right  circular  cone  whose  vertical  angle 
is  a  right  angle,  hence  the  polar  plane  of  any  one  of  these  rays  is 
tangent  to  the  cone  along  a  line  diametrically  opposite  to  the  ray. 
The  polarity  in  the  plane,  then,  is  such  that  the  polar  line  of  any 
point  D  on  the  circle  is  the  tangent  at  the  opposite  end  of  the 
diameter  through  D.  The  polar  line  of  any  point  as  A  is  a  line 
a  perpendicular  to  the  diameter  AC  at  a  point  M  such  that 

AC-CM  =  CO2  =  r\ 

It  follows  that  CM  =  CM';  the  two  polar  lines  a  and  a'  cut  the 
diameter  AC  at  points  equidistant  from  the  center  of  the  circle 
and  on  opposite  sides  of  it.  On  account  of  this  property,  any  sec- 
tion of  the  orthogonal  polarity  in  a  bundle  is  called  an  anti- 
polarity  with  respect  to  a  circle,  the  radius  of  the  circle  being  equal 


192  PROJECTIVE  GEOMETRY  [§155 

in  length  to  the  perpendicular  from  the  center  of  the  bundle  upon 
the  cutting  plane. 

In  an  anti-polarity  with  respect  to  a  circle,  the  polar  line  of  the 
center  of  the  circle  is  the  ideal  line,  the  involution  of  conjugate 
lines  about  the  center  is  circular,  and  the  involution  of  conjugate 
points  on  the  ideal  line  coincides  with  the  involution  determined 
upon  this  line  by  the  circle. 

155.  The  Absolute  Polarity. — The  sections  of  all  the  orthogonal 
polarities  in  space  by  the  ideal  plane  coincide  in  one  and  the  same 
polarity  in  the  ideal  plane. 

For,  suppose  p  and  p'  are  parallel  lines  in  two  orthogonal  polari- 
ties. The  polar  planes,  TT  and  TT'  are  necessarily  parallel  to  each 
other;  hence  the  two  pairs  of  elements  p,  TT  and  p',  TC'  are  cut  by 
the  ideal  plane  in  the  same  pair  of  elements,  and  these  are  pole  and 
polar  line  in  the  polarity  in  the  ideal  plane.  This  polarity  is  called 
the  absolute  polarity.  It  is  uniform  and  defines  the  imaginary 
circle  at  infinity. 

Exercises 

1.  Show  how  to  construct  a  uniform  polarity  in  a  plane. 

2.  Construct  five  points  of  the  conic  which  is  the  locus  of  self -con- 
jugate points  in  a  given  non-uniform  polarity. 

3.  The  section  of  an  orthogonal  polarity,  by  a  plane  passing  through 
the  center  of  the  bundle,  consists  of  pairs  of  conjugate  rays  forming  a 
circular  involution. 

4.  In  a  given  polarity  in  a  plane,  show  that  two  polar  triangles 
having  no  elements  in  common  are  in  perspective  position.     Use  Art. 
151,  exercise  9. 

6.  A  conic  can  be  circumscribed  about  two  polar  triangles  in  a  plane. 
The  sides  of  the  triangles  touch  another  conic. 

156.  Double  Polarities  in  a  Plane  and  in  a  Bundle. — If  two 

polarities  exist  together  in  a  plane  or  in  a  bundle,  we  shall  speak 
of  them  as  forming  a  double  polarity  in  the  plane  or  in  the  bundle. 
Let  PI  and  PI  denote  two  polarities  forming  a  double  polarity  in 
a  plane.  In  general,  any  line  a  in  the  plane  has  two  poles  A\  and 
A «,  one  in  each  polarity;  a  triangle  whose  sides  are  a,  b,  c,  has  two 
polar  triangles  Ai,  BI,  Ci  and  A2,  B2,  C2,  one  in  each  polarity;  a 
sheaf  of  rays  has  two  point-rows  of  poles  which  are  projectively 
related  to  each  other,  since  each  is  projectively  related  to  the  sheaf 


§157]    POLARITIES  IN  A  PLANE  AND  IN  A  BUNDLE     193 

of  rays.  Any  point  A  in  the  plane  has,  in  general,  two  polar  lines 
d  and  a2,  one  in  each  polarity;  a  point-row  has  two  sheaves  of 
polar  lines  which  are  projectively  related  to  each  other,  since  each 
is  projectively  related  to  the  point-row.  It  follows  that  there  is  a 
collineation  in  the  plane  in  which  two  corresponding  points  are  the 
poles  of  the  same  line,  and  two  corresponding  lines  are  the  polar 
lines  of  the  same  po'int. 

If  we  regard  a  polarity  as  an  operation  which  transforms  all  the 
elements  of  a  plane  into  their  respective  polar  elements,  we  can 
say  that  the  collineation,  whose  existence  has  just  been  shown,  is 
the  result  of  performing  the  operations  PI  and  P2,  one  after  the 
other.  For,  if  A\  and  Az  are  the  poles  of  a  in  Pi  and  P2,  respec- 
tively, then  PI  changes  A  i  into  a,  and  P2  changes  a  into  A%.  Sym- 
bolically expressed, 

Pi(Ai)  =  a, 

P2(a)     =  At, 
and  hence, 

Pp  ( A   \    p  (n\   4 
1L\\**-\)     —    fjKftf     —    -il2. 

For  convenience,  we  shall  denote  the  resultant  collineation  of  the 
two  polarities  by  the  letter  R.  We  can  then  write, 

P.P!  =  R. 

In  the  same  way,  two  polarities  in  a  bundle  have  a  resultant 
collineation  in  which  any  two  corresponding  planes  are  the  polar 
planes  of  the  same  ray,  and  any  two  corresponding  rays  are  the 
pole-rays  of  the  same  plane.  We  shall  also  denote  this  collineation 
by  the  letter  R. 

157.  Common  Elements  in  a  Double  Polarity. — A  point  and  a 
line  are  common  elements  in  a  double  polarity  in  a  plane  if  they 
are  pole  and  polar  line  in  both  polarities;  a  ray  and  a  plane  are 
common  elements  in  a  double  polarity  in  a  bundle  if  they  are  pole- 
ray  and  polar  plane  in  both  polarities. 

The  common  elements  in  a  double  polarity  (P2,  PI)  are  the  self- 
corresponding  elements  of  the  resultant  collineation  R,  and  conversely. 

For,  if  P,(A)  =  a  and  P2(a)  =  A,  then  PzPi(A)  =  P2(a)  =A, 
or  R(A)  =  A.  Similarly,  R  (a)  =  a.  Thus  the  common  elements, 
A,  a,  in  (P2,  PI)  are  self-corresponding  elements  in  R. 

Conversely,  if  PzP^A)  —  A  and  P\(A)  =  a,  then  P2(a)  =  A, 

13 


194  *    PROJECTIVE  GEOMETRY  [§158 

PI  (a)  =  A,    and  P2(A)  =  a.     Thus  self-corresponding  elements 
of  R  are  common  elements  in  (P2,  -Pi). 
There  are  two  cases  to  consider: 

1.  If  R  is  a  perspectivity,  then,  in  a  plane,  (P%,  PI)  has  a  com- 
mon point-row  of  poles  along  the  axis  of  perspectivity  and  a  com- 
mon sheaf  of  polar  lines  about  the  center  of  perspectivity.     The 
center  and  the  axis  are  pole  and  polar  line  in  each  polarity. 

As  an  example,  the  double  polarity  formed  by  the  poles  and 
polar  lines  with  respect  to  two  concentric  circles  in  a  plane  has  a 
resultant  collineation  which  is  a  perspectivity.  The  axis  of  this 
perspectivity  is  the  ideal  line,  and  the  center  is  the  common 
center  of  the  two  circles. 

In  a  bundle,  (Pz,  PI)  has  a  common  sheaf  of  pole-rays  and  a 
common  sheaf  of  polar  planes.  The  axis  of  the  sheaf  of  planes  and 
the  plane  of  the  sheaf  of  rays  are  pole-ray  and  polar  plane  in  each 
polarity  (Art.  133). 

2.  If  R  is  not  a  perspectivity,  then  (P2,  PI)  in  a  plane  has  at  least 
one  common  point  and  one  common  line  which  are  pole  and  polar 
line  in  both  polarities,  or  at  most  the  vertices  and  sides  of  a 
common  self-polar  triangle.     For  example,  the  double  polarity 
formed  by  the  poles  and  polar  lines  with  respect  to  two  conies, 
having  four  common  points,  has  the  vertices  and  sides  of  the 
common  self-polar  triangle  as  common  elements. 

In  a  bundle,  (P2,  PI)  has  at  least  one  common  ray  and  one 
common  plane  which  are  pole-ray  and  polar  plane  in  both  po- 
larities, or  at  most  the  edges  and  faces  of  a  common  self-polar 
pyramid  (Art.  147,3). 

158.  Double  Conjugate  Elements. — In  a  double  polarity  in  a 
plane,  any  point  as  A  (Fig.  115)  has,  in  general,  two  polar  lines 
0,1  and  a2  which  intersect  in  a  point  B.  The  polar  lines  of  B  inter- 
sect in  A .  The  points  A  and5  are  conjugate  to  each  other  in  both 
polarities  and  shall  be  called  double  conjugate  points.  Recipro- 
cally, two  lines  a  and  b  are  double  conjugate  lines,  if  they  are 
conjugate  lines  in  both  polarities,  that  is,  if  each  joins  the  poles  of 
the  other. 

In  a  bundle,  two  rays  are  double  conjugate  rays,  if  they  are 
conjugate  rays  in  both  polarities;  and  two  planes  are  double  con- 
jugate planes,  if  they  are  conjugate  planes  in  both  polarities. 


§158]    POLARITIES  IN  A  PLANE  AND  IN  A  BUNDLE     195 


If,  in  Fig.  115,  A  describes  the  point-row  s,  B  will,  in  general, 
describe  a  conic  k.  For  the  polar  lines  of  A  will  describe  sheaves 
about  the  poles  Si  and  S2  of  s  which  are  projectively  related  to  each 
other,  since  each  is  projectively  related  to  the  point-row  described 
by  A.  These  sheaves  generate  the  conic  k.  The  line  s  and  the 
conic  k  correspond  to  each 
other  point  to  point.  Each  is 
the  locus  of  double  conjugates 
to  the  points  of  the  other. 

If,  however,  s  passes  through 
a  common  point  P  of  the  double 
polarity  (Fig.  116),  the  sheaves 
Si  and  Si  are  in  perspective  posi- 
tion, since  the  polar  lines  of  P 
coincide  in  the  line  S\  Sz.  The  locus  of  double  conjugates  to  the 
points  of  s  is  then  a  line  t  which  also  passes  through  P,  since 
the  polar  lines  of  R  pass  through  P.  The  lines  s  and  t  are  cor- 
responding loci  of  double  conjugate  points. 

Reciprocally,  if  a  and  b  are  double  conjugate  lines  in  a  plane  and 
a  describes  a  sheaf  of  rays  about  a  point  S,  b,  will,  in  general, 


describe  an  envelope  of  second  class  K  whose  rays  join  correspond- 
ing poles  on  the  polar  lines  Si  and  s2  of  S.  But  if  S  lies  on  a  com- 
mon line  of  the  double  polarity,  the  two  point-rows  of  poles  cor- 
responding to  rays  of  S  are  in  perspective  position,  and  b  will  then 
describe  a  sheaf  of  rays  of  first  class  about  a  point  T  also  on  the  same 
common  line. 

Similar    conclusions    hold    concerning    corresponding    loci    of 
double  conjugate  elements  in  a  bundle. 


196  PROJECTIVE  GEOMETRY  [§159 

159.  Confocal  Elements   in   a  Double   Polarity. — When   two 
polarities  are  in  the  same  plane,  there  are  two  involutions  of 
conjugate  points  along  each  line  of  the  plane,  and  two  involutions 
of  conjugate  rays  about  each  point  of  the  plane.     Any  line  upon 
which  the  two  involutions  coincide  is  a  confocal  line  of  the  double 
polarity,  and  any  point  about  which  the  two  involutions  coincide 
is  a  confocal  point  of  the  double  polarity. 

Similarly,  when  two  polarities  are  in  the  same  bundle,  there  are 
two  involutions  of  conjugate  rays  in  each  plane  of  the  bundle,  and 
two  involutions  of  conjugate  planes  about  each  ray  of  the  bundle. 
Any  plane  in  which  the  two  involutions  coincide  is  a  confocal 
plane  of  the  double  polarity,  and  any  line  about  which  the  two 
involutions  coincide  is  a  confocal  axis  of  the  double  polarity. 

160.  Construction   of   the    Confocal   Elements    of   a   Double 
Polarity. — 

1.  If  the  resultant  collineation  P-fi  =  R  is  a  perspectivity, 
the  polar  lines  of  any  point  A  of  its  axis  coincide  and  intersect  the 
axis  in  the  double  conjugate  point  B  (Fig.  117).     Reciprocally, 
the  poles  of  any  line  a  through  the  center  coincide  and  determine 

the  double  conjugate  ray  b.  Consequently,  the 
involutions  determined  by  PI  and  P2  along  the 
axis  and  about  the  center  coincide,  element  to 
element.  Hence  the  axis  of  R  is  a  confocal  line 
and  the  center  of  R  is  a  confocal  point.  It  is  easy 
to  see  that  these  are  the  only  confocal  elements  in 
the  plane. 

In  a  bundle,  the  involutions  about  the  axis  of 
the  common  sheaf  of  planes  coincide,  and  the  involutions  in  the 
plane  of  the  common  sheaf  of  rays  coincide.  These  are  the  only 
confocal  elements  in  the  bundle. 

2.  If  R  is  not  a  perspectivity,  we  shall  suppose  that  PI  and  P2 
have  a  common  self -polar  triangle  ABC  (Fig.  118).    Any  line 
through  a  vertex  of  this  triangle  has  for  its  corresponding  locus 
of  double  con  jugate 'points  a  line  through  the  same  vertex  (Art. 
158).     Thus,  if  S  and  T  are  double  conjugate  points,  the  lines 
joining  S  and  T  to  the  vertices  A,  B,  C  are  pairs  of  corresponding 
loci  of  double  conjugate  points.     If,  for  example,  S  describes  the 
line  /,  T  described  the  line  n,  and  conversely.     The  two  point- 


§160]    POLARITIES  IN  A  PLANE  AND  IN  A  BUNDLE     197 

rows,  I  and  n,  are  projec lively  related  and  are  projected  from  B, 
say,  in  projectively  related  sheaves  of  rays  described  respectively 
by  the  rays  s  and  t.  These  sheaves  are  not  only  projectively  re- 
lated, but  are  in  involution,  since  a  pair  of  rays  is  a  pair  of  corre- 
sponding loci  of  double  conjugate  points.  Hence: 

About  each  vertex  of  the  self -polar  triangle  there  is  an  involution 
such  that  to  any  ray  corresponds  its  locus  of  double  conjugate  points. 

The  three  involutions  will  be  all  hyperbolic  if  any  pair  of  double 
conjugate  points,  as  S  and  T,  occupy  the  same  region  of  the  plane 


defined  by  the  self-polar  triangle  ABC.  If  S  and  T  occupy 
different  regions,  two  of  the  involutions  are  elliptic  and  one  is 
hyperbolic. 

The  focal  rays  in  any  one  of  these  involutions  are  lines  each  of 
which  is  its  own  locus  of  double  conjugate  points,  that  is,  they  are 
confocal  lines.  We  conclude  therefore: 

There  are  either  six  real  confocal  lines  intersecting  in  pairs  in 
the  vertices  of  the  common  self-polar  triangle,  or  else  there  are  two 
real  confocal  lines  meeting  in  one  vertex  of  the  common  self-polar 
triangle. 

The  involutions  which  thus  determine  the  confocal  lines  are 
known  as  soon  as  one  pair  of  double  conjugate  points  is  known. 
For  two  sides  of  the  self-polar  triangle  form  a  pair  of  rays  in  one 
of  these  involutions,  since  the  polar  lines  of  a  point  on  any  side 
intersect  in  the  opposite  vertex.  The  involution  about  the 
vertex  B,  for  example,  is  determined  by  the  pairs  TB,  SB  and  AB, 
CB. 

Reciprocally  we  have  the  following  statements: 

On  each  side  of  the  common  self-polar  triangle  there  is  an  involution 
such  that  to  any  point  S  Corresponds  the  center  T  of  its  sheaf  of  double 


198  PROJECTIVE  GEOMETRY  (§160 

cm/jugate  rays.  The  two  vertices  of  the  triangle  on  the  same  side  are 
corresponding  points  in  the  involution  on  that  side.  One  of  these 
involutions  is  always  hyperbolic;  the  other  two  may  be  hyperbolic 
or  elliptic.  There  are,  therefore,  either  six  real  confocal  points 
lying  by  pairs  on  the  sides  of  che  self-polar  triangle,  or  else  there  are 
two  real  confocal  points  lying  on  one  side  of  the  self-polar  triangle. 

We  see  at  once  by  projection  that,  if  a  double  polarity  in  a 
bundle  has  a  common  self-polar  pyramid: 

A  double  polarity  in  a  bundle  has  either  six  real  confocal  planes 
intersecting  by  pairs  in  the  edges  of  the  common  self-polar  pyramid, 
or  else  it  has  two  real  confocal  planes  meeting  in  one  edge  of  the  com- 
mon self-polar  pyramid;  and  it  has  either  six  real  confocal  axes  lying 
by  pairs  in  the  faces  of  the  common  self-polar  pyramid,  or  else  it 
has  two  real  confocal  axes  lying  in  one  face  of  the  common  self-polar 
pyramid. 

Exercises 

1.  A  double  polarity  in  a  plane  is  determined  by  two  real  conies 
which  intersect  in  four  real  points  A,  B,  C,  D. 

(a)  The  polarities  have  a  real  common  self-polar  triangle  which  is 
the  diagonal  triangle  of  the  quadrangle  A  BCD. 

(b)  The  six  sides  of  the  quadrangle  ABCD  are  confocal  lines. 

(c)  The  six  vertices  of  the  quadrilateral  whose  sides  are  the  common 
tangents  to  the  conies  are  confocal  points. 

2.  A  double  polarity  is  determined  by  two  confocal  conies. 

(a)  The  common  self-polar  triangle  consists  of  the  axes  and  the 
ideal  line. 

(b)  The  foci  are  the  real   confocal   points.     Common  chards  are 
real  confocal  lines. 

3.  A  given  circle  determines  a  polarity  and  an  anti-polarity  in  its 
plane.     Show  that  the  resultant  collineation  of  the  double  polarity  in 
the  plane  is  a  perspectivity. 

4.  A  double  polarity  is  determined  by  two  circles  in  a  plane  which 
are  not  concentric. 

(a)  Construct  the  common  self-polar  triangle. 

(6)  The  centers  of  similitude  are  confocal  points.  Are  there  more 
than  two  real  confocal  points? 

(c)  Determine  the  number  and  position  of  the  real  confocal  lines. 

5.  A  circle  is  concentric  with  an  ellipse.     Show  that  the  anti-polar- 


§161]    POLARITIES  IN  A  PLANE  AND  IN  A  BUNDLE     199 


ity  with  respect  to  the  circle  and  the  polarity  with  respect  to  the  ellipse 
have  in  common  a  real  self-polar  triangle  and  that  there  are  two  real 
confocal  lines  and  two  real  confocal  points. 

161.  Application  to  Cones  of  the  Second  Order. — A  cone  of 
second  order  sets  up  a  polarity  in  the  bundle  to  which  it  belongs 
such  that  corresponding  elements  are  pole-ray  and  polar  plane 
with  respect  to  the  cone.  This  polarity,  which  we  shall  denote 
by  PI,  is  non-uniform;  its  self-conjugate  lines  are  rays  of  the  cone 
and  its  self-conjugate  planes  are  tangent  planes  to  the  cone. 

We  shall  now  consider  the  orthogonal  polarity  P2  in  the  same 
bundle.  The  double  polarity  (P2,  PI)  has  a  resultant  collineation 
R;  and  there  are  two  cases  to  consider  according  as  R  is  a  perspec- 
tivity  or  not  a  perspectivity. 

1.  If  R  is  a  perspectivity,  then  PI  and  P2  have  in  common  a 
sheaf  of  rays  and  a  sheaf  of  planes  (Art.  157,  1)  such  that  the  plane 
a  of  the  sheaf  of  rays  and  the  axis  a  of 
the  sheaf  of  planes  are  polar  plane  and 
pole-ray  in  both  polarities;  that  is, 
the  plane  a  is  perpendicular  to  the 
line  a,  .since  P2  is  the  orthogonal 
polarity  (Fig.  119).  Moreover,  a  is 
the  confocal  plane  and  a  is  the  confocal 
axis,  and  the  coinciding  involutions 
are  circular  (Art.  160,  1).  Conse- 
quently, if  we  cut  the  cone  by  plane 
parallel  to  a,  and  which  cuts  the 
line  a  in  the  point  A,  we  obtain  a 
conic  such  that  the  involution  of  con- 


FIG.  119. 


jugate  rays  about  A  is  circular.  Hence,  A  is  a  focus  of  the 
conic.  But  since  the  directrix  (the  intersection  of  a  with  the 
plane  of  the  conic)  is  infinitely  distant,  the  conic  is  a  circle. 
Hence,  the  cone  is  a  right  circular  cone  whose  axis  is  a. 

2.  If  R  is  not  a  perspectivity,  PI  and  P2  have  in  common  a  real 
self-polar  pyramid. 

For  the  two  polarities  have  in  common  at  least  one  line  a 
and  one  plane  a  which  are  pole-ray  and  polar  plane  in  both  polari- 
ties. Hence  a  and  «  are  perpendicular  to  each  other.  In  the 
plane  a  there  are  two  involutions  of  conjugate  rays,  one  for  each 


200 


PROJECTIVE  GEOMETRY 


[§161 


polarity.  The  involution  belonging  to  P%  is  necessarily  elliptic, 
and  hence  the  two  involutions  have  two  real  rays  in  common 
(Art.  117,  3)  which  we  will  call  b  and  c.  These  rays,  being  con- 


FIG.  120. 

jugate  in  the  polarity  P2,  are  perpendicular  to  each  other  and  each 
is  perpendicular  to  a,  since  they  lie  in  the  plane  a.  The  three 
rays  a,  b,  c  form  the  edges  of  the  common  self-polar  pyramid  and 
are  the  principal  axes  of  the  cone.  The  faces  of  the  self-polar 
pyramid  are  the  diametral  planes  of  the  cone. 


§162]    POLARITIES  IN  A  PLANE  AND  IN  A  BUNDLE     201 

One  of  the  axes  must  be  within  the  cone,  since  of  the  involutions 
about  the  axes  which  belong  to  PI,  one  is  necessarily  elliptic  (Art. 
151,  second  case).  This  is  the  major  axis  of  the  cone  and  is  de- 
noted by  the  letter  a  (Fig.  120).  Any  section  of  the  cone  by  a  plane 
perpendicular  to  a  is  an  ellipse  whose  axes  are  parallel  to  the  other 
two  axes  of  the  cone.  That  axis  of  the  cone  which  is  parallel 
to  the  major  axis  of  the  ellipse  is  the  mean  axis  of  the  cone  and  is 
denoted  by  the  letter  b.  The  third  axis  is  the  minor  axis  of  the 
cone  and  is  denoted  by  the  letter  c.  Any  section  of  the  cone 
perpendicular  to  either  6  or  c  is  a  hyperbola. 

162.  Cyclic  Planes  and  Focal  Axes  of  Cones. — //  the  resultant 
collineation  R  =  PzPi  is  not  a  perspectivity ,  there  are  two  real 
confocal  planes  intersecting  in  the  mean  axis  of  the  cone  and  two 
real  confocal  axes  in  the  diametral  plane  perpendicular  to  the  minor 
axis  of  the  cone. 

For,  if  we  cut  the  double  polarity  by  a  plane  perpendicular  to  the 
major  axis  of  the  cone,  we  obtain  a  double  polarity  in  the  cutting 
plane  composed  of  a  polarity  with  respect  to  the  ellipse  (section 
of  the  cone)  and  an  anti-polarity  with  respect  to  a  circle  concentric 
with  the  ellipse  (Art.  154).  This  double  polarity  has  a  real 
self-polar  triangle  whose  sides  are  the  axes  of  the  ellipse  and  the 
ideal  line.  There  are  but  two  real  confocal  lines,  and  these  are 
parallel  to  the  major  axis  of  the  ellipse.  Also  there  are  but  two 
real  confocal  points,  and  these  lie  on  the  major  axis  of  the  ellipse 
(cf.  Art.  160,  exercise  5).  These  confocal  elements  are  the  traces 
of  the  confocal  elements  of  the  double  polarity  in  the  bundle. 

The  confocal  planes  of  the  double  polarity  are  the  cyclic  planes 
of  the  cone ;  and  the  confocal  axes  of  the  double  polarity  are  the 
focal  axes  of  the  cone. 

Any  section  of  a  cone  parallel  to  a  cyclic  plane  is  a  circle. 

For,  if  a  is  a  cyclic  plane  and  SD  is  any  ray  in  a  through  S, 
the  conjugate  ray  SD'  is  perpendicular  to  SD,  since  the  coinciding 
involutions  in  a  are  circular.  The  polar  planes  of  SD  and  of  SD' 
in  PI  are  cut  by  any  plane  parallel  to  a  in  a  pair  of  conjugate 
diameters  of  the  conic  k',  cut  from  the  cone  (Fig.  120).  It 
follows  that  any  two  conjugate  diameters  of  k'  are  mutually  per- 
pendicular, and  hence  k'  is  a  circle. 

Any  plane  perpendicular  to  a  focal  axis  of  a  cone  cuts  the  cone, 


PROJECTIVE  GEOMETRY  [§163 

in  a  conic  k"  and  the  focal  axis  in  a  point  F  which  is  one 
focus  of  k". 

For  the  coinciding  involutions  about  the  focal  axid  are  circular. 
Conjugate  planes  in  PI  are^  therefore,  cut  in  normal  conjugate 
rays  with  respect  to  the  conic  k".  F  is  therefore  a  focus  of  k". 

163.  Quadric  Transformations. — When  two  polarities  exist  in 
the  same  plane,  the  transformation  which  changes  every  point 
of  the  plane  into  its  double  conjugate  point  is  called  a  quadric 
transformation.  We  have  already  noticed  the  particular  quadric 
transformation  that  arises  from  non-uniform  polarities  in  the 
same  plane  (Art.  72,  exercise  6). 

A  quadric  transformation  is  not  a  collineation,  since  any  point- 
row  s  corresponds,  in  general,  to  a  conic  k  which  is  the  locus  of  double 
conjugates  to  the  points  of  s  (Art.  158). 

The  nature  of  a  quadric  transformation  depends  upon  the  two 
polarities  from  which  it  arises.  If  PI  and  Pz  are  two  polarities 
and  R,  their  resultant  collineation,  is  a  perspectivity,  then  the 
double  conjugate  of  any  point  in  the  plane  is  on  the  axis  of  per- 
spectivity, since  a  pair  of  corresponding  polar  lines  must  meet  on 
the  axis.  In  this  case  the  quadric  transformation  transforms 
every  line  in  the  plane  into  the  axis  of  perspectivity,  and  is,  for 
this  reason,  called  degenerate  or  trivial. 

If  R  is  not  a  perspectivity,  the  self-corresponding  elements  of  R 
are  called  the  fundamental  elements  of  the  quadric  transformation. 
Thus,  if  PI  and  P2  have  in  common  the  vertices  and  sides  of  a 
self-polar  triangle,  then  the  vertices  of  this  triangle  are  the  funda- 
mental points,  and  the  sides  are  the  fundamental  lines,  of  the 
quadric  transformation  arising  from  the  double  polarity  (P2,  PI). 
A  vertex  of  this  triangle  corresponds  to  any  point  on  the  opposite 
side,  since  the  polar  lines  of  a  vertex  coincide  in  the  opposite 
side;  and  any  point  on  a  side  of  this  triangle  corresponds  to  the 
opposite  vertex,  since  the  polar  lines  of  the  point  intersect  in  the 
opposite  vertex. 

1.  Any  line  s,  not  passing  through  any  fundamental  point  of  a 
quadric  transformation,  corresponds  to  a  conic  k  containing  all  the 
fundamental  points. 

For  the  line  s  meets  the  fundamental  lines  of  the  quadric  trans- 
formation in  points  whose  double  conjugates  are  the  fundamental 


§104]    POLARITIES  IN  A  PLANE  AND  IN  A  BUNDLE     203 

points  of  the  quadric  transformation.     Therefore,  the  conic  k 
passes  through  all  the  fundamental  points. 

2.  A  line  s  passing  through  a  fundamental  point  of  a  quadric 
transformation  corresponds  to  a  line  t  passing  through  the  same 
fundamental  point. 

For  the  locus  of  double  conjugates  to  the  points  of  s  is  a  line 
t  through  the  same  fundamental  point  (Art.  158). 

3.  A  conic  k  passing  through  two  of  the  fundamental  points  of  a 
quadric  transformation  corresponds  to  a  conic  k'  passing  through 
the  same  two  fundamental  points. 

Let  A  and  B  be  two  fundamental  points  and  P,  any  point  of  the 
conic  k  (Fig.  121).  The  lines  s  and  s',  joining  P  to  A  and  B, 
correspond  in  the  quadric  trans- 
formation to  lines  t  and  t'  inter- 
secting in  a  point  Q.  P  and  Q 
are  corresponding  points  in  the 
quadric  transformation.  As  P 
described  the  conic  k,  the  lines 
,9  and  s'  describe  projectively 
related  sheaves  of  rays  about  A 
and  B.  But  the  sheaf  described 
by  s  is  projectively  related  to 
the  sheaf  described  by  t,  and  the  JTIG 

sheaf  described  by  s'  is  project- 
ively related  to  the  sheaf  described  by  t'  (Art.  160,2).    Hence, 
the  sheaves  described  by  t  and  t'  are  projectively  related  and 
generate  a  conic  k',  the  locus  of  Q. 

164.  Perspective  Quadric  Transformations. — Let  A  be  the 
center  of  a  sheaf  of  rays  of  first  class  and  PI,  a  polarity  in  the  same 
plane  with  the  sheaf  of  rays. 

The  transformation  which  changes  any  point  P  into  the  point 
Q,  the  intersection  of  PA  with  the  polar  line  of  P,  is  a  quadric 
transformation. 

For,  if  P  describes  a  point-row  s  (Fig.  122),  the  line  PA  describes 
the  sheaf  A  and  the  polar  line  of  P  describes  the  sheaf  of  polar 
lines  about  the  pole  S  of  s.  Hence, 

sheaf  A  A"  point-row  s  A  sheaf  S, 


FIG.  122. 


204  PROJECTIVE  GEOMETRY  [§165 

and  therefore, 

sheaf  A  A  sheaf  S. 

The  intersection  of  corresponding  rays  in  these  sheaves  describes 

a  conic  k  ,  the  locus  of  Q.  Hence, 
in  general,  a  point-row  s  corre- 
sponds to  a  conic  k. 

The  transformation  which 
changes  P  into  Q  is  involutoric, 
since  the  polar  line  of  Q  passes 
through  P. 

Any  line  passing  through  A 
corresponds  to  itself  but  not 
point  to  point.  A  is  a  funda- 
mental point  of  the  quadric  transformation.  The  polar  line  of  A 
in  PI  is  a  fundamental  line  of  the  quadric  transformation,  since 
the  polar  line  of  any  point  on  this  line  must  pass  through  A. 

If  the  polarity  Pi  is  non-uniform,  it  has  a  conic  of  self-conjugate 
points  k'.  Any  point  on  A;'  is  a  self-corresponding  point  of  the 
quadric  transformation,  since  it  lies  on  its  polar  line.  Two  corre- 
sponding points  of  the  quadric  transformation  are  collinear  with  A 
and  harmonically  separated  by 
k'.  If  Pi  is  uniform,  there  are 
no  real  self-corresponding  points 
of  the  quadric  transformation. 

165.  Inversion  with  Respect 
to  a  Circle. — Two  important 
perspective  quadric  transforma- 
tions are  the  inversions  with  re- 
spect to  a  fixed  circle.  When 
the  polarity  PI  is  either  a  polarity 
or  an  antipolarity  with  respect 
to  a  circle  whose  center  is  also 


FIG.  123. 


the  center  of  the  sheaf  A,  the  perspective  quadric  transformation 
is  called  an  inversion  with  respect  to  the  circle. 

Suppose  that  PI  is  a  polarity  with  respect  to  a  fixed  circle  whose 
radius  is  r  and  whose  center  is  A  (Fig.  123).  Let  s  be  any  line  not 
passing  through  A.  As  P  describes  s,  its  polar  line  describes  the 


§166]    POLARITIES  IN  A  PLANE  AND  IN  A  BUNDLE    205 

pro j  actively  related  sheaf  of  lines  about  the  pole  S,  and  the  line  AP 
describes  the  sheaf  A  in  perspective  position  with  s.  The  sheaves 
S  and  A  generate  a  circle  on  SA  as  diameter.  This  circle  corre- 
sponds to  s  in  the  inversion.  P  and  PI  are  corresponding  points  in 
the  inversion. 

Since  PI  is  on  the  polar  line  of  P  with  respect  to  the  fixed  circle, 
we  have  (Art.  154), 

AP-APi  =  r\ 

On  account  of  this  relation,  an  inversion  is  often  called  a  transfor- 
mation by  reciprocal  radii. 

If  PI  is  an  anti-polarity  with  respect  to  the  fixed  circle,  the  point 
P  corresponds  to  the  point  P'i,  the  intersection  of  AP  with  the  anti- 
polar  line  of  P.  The  line  s  corresponds  to  the  circle  on  S'A  as 
diameter.  In  this  case  we  have, 

APAP\  =  -r\ 

the  negative  sign  being  used  because  AP  and  AP'i  are  measured 
in  opposite  directions. 

In  either  of  the  inversions,  the  center  of  the  fixed  circle  is  called 
the  center  of  inversion.  We  may  distinguish  between  the  two  in- 
versions by  calling  the  one  positive,  and  the  other  negative.  The 
center  of  inversion  is  the  fundamental  point,  and  the  ideal  line  is 
the  fundamental  line,  of  either  inversion.  Every  point  of  the 
fixed  circle  is  a  self-corresponding  point  of  a  positive  inversion. 
There  are  no  self-corresponding  points  of  a  negative  inversion. 

166.  Properties  of  Positive  Inversions. — 

1.  To  a  circle  not.  passing  through  the  center  of  inversion  corre- 
sponds a  circle  not  passing  through  the  center  of  inversion. 

For,  let  0  be  the  center  of  any  circle  k  not  passing  through  A 
(Fig.  124),  and  let  AO  meet  k  in  S  and  T.    Let  P  be  any  point  on 
k,  and  construct  the  points  corresponding  to  S,  T,  P  in  the  inver- 
sion.   Let  these  be  the  points  Si,  TI,  PI.    Then  we  have, 
AS-ASl  =  AP-APl  =  r2,  or, 

ASi-.AP.r.AP-.AS. 

Hence  the  triangles  ASiPi  and  APS  are  similar.  Likewise,  the 
triangles  A  T\Pi  and  APT  are  similar.  It  follows  from  elementary 
geometry  that, 


206  PROJECTIVE  GEOMETRY  [§166 

Therefore,  as  P  describes  the  circle  k,  PI  describes  a  circle  kt  on 
SiTi  as  diameter.  If  k  meets  the  fixed  circle  in  real  points,  ki 
must  pass  through  these  points. 

2.  The  center  of  inversion  is  a  center  of  similitude  of  any  pair  of 
corresponding  circles  k  and  k\.     Corresponding  points  on  these  circles 
are  inverse  points  with  respect  to  this  center  of  similitude  (Art.  145). 

3.  Any  circle  passing  through  a  pair  of  corresponding  points  is 
inverse  to  itself,  that  is,  corresponds  to  itself  in  the  inversion. 


FIG.  124. 


4.  An  inversion  transforms  every  point  outside  the  fixed  circle  into 
a  point  inside  the  fixed  circle,  and  vice  versa. 

5.  Any  two  straight  lines  of  the  plane  intersect  in  the  same  angle 
as  do  their  corresponding  circles. 

For,  if  we  draw  lines  through  the  center  of  inversion  and  parallel 
to  the  given  lines,  these  will  touch  the  corresponding  circles  at  the 
center  of  inversion.  Consequently,  corresponding  angles  of  two 
inverse  figures  are  equal. 

Any  transformation  which  does  not  change  the  magnitude  of 
the  angles  of  a  figure  is  called  a  conform  transformation.  The 
inversions  are  conform  transformations. 


§167]    POLARITIES  IN  A  PLANE  AND  IN  A  BUNDLE     207 

Exercises 

1.  A  perspective  quadric   transformation  arises  from   a  polarity 
with  respect  to  a  fixed  conic  and  a  given  sheaf  of  lines  A.     Show  that 
the  conic  corresponding  to  any  line  not  passing  through  A  meets  the 
fixed  conic  in  the  points  where  it  is  met  by  the  polar  line  of  A. 

2.  In  an  inversion,  a  system  of  concentric  circles  corresponds  to  a 
system  of  circles  with  a  common  radical  axis. 

3.  If  two  circles  touch  each  other,  their  inverse  circles  also  touch 
each  other. 

4.  Show    that    a    circle    inverse    to    itself   cuts   the   fixed   circle 
orthogonally. 

6.  Show  how  to  invert  two  given  circles  into  two  equal  circles. 

6.  Do  the  properties  given  in  Art.  166  hold  for  negative  inversions? 

167.  Circular  Transformations. — If  a  plane  is  first  transformed 
into  itself  by  a  similitude  S  and  then  inverted  with  respect  to  a 
fixed  circle  by  a  positive  inversion  I,  the  resultant  transformation, 

C  =  I-S, 
is  called  a  circular  transformation. 

Any  line  in  the  plane  can  be  regarded  as  a  circle  of  infinite  radius. 
With  this  proviso,  we  can  say  that: 

1.  A  circular  transformation  always  changes  circles  into  circles. 
For  the  similitude  S  changes  circles  into  circles  (Art.  144,  2)  and 
the  inversion  /  also  changes  circles  into  circles. 

2.  Circular  transformations  are  conform.     For  neither  the  simil- 
itude nor  the  inversion  alters  the  angles  of  a  figure. 

Quadric  transformations  arising  from  two  polarities  in  a  plane 
and  perspective  quadric  transformations  have  in  common  two  im- 
portant properties.  First,  they  are  one  to  one  "  point- transforma- 
tions"; that  is,  to  any  point  corresponds,  in  general,  but  a  single 
point.  The  exceptions  are  the  fundamental  points.  Any  trans- 
formation having  this  property  is  called  birational.  Second,  they 
are  involutoric  in  character;  that  is,  the  points  of  any  pair  corre- 
spond to  each  other  doubly. 

A  circular  transformation  is  birational  but  is  not,  in  general, 
involutoric. 

3.  Any  birational  transformation  not  a  similitude  which  changes 
circles  into  circles  is  a  circular  transformation  or  an  inversion. 

Let  C  be  the  transformation  in  question.     In  the  first  place,  C 


208  PROJECTIVE  GEOMETRY  [§168 

must  have  a  fundamental  point  A  through  which  pass  all  the  circles 
corresponding  to  the  straight  lines  of  the  plane.  For  not  all  the 
lines  of  the  plane  can  be  changed  into  lines,  since  C  is  not  a  simili- 
tude. Consider  two  lines  a  and  b;  their  corresponding  circles  meet 
in  two  points,  only  one  of  which  can  correspond  to  the  intersection 
of  a  and  6,  since  C  is  birational.  The  other  must  be  a  fundamental 
point  A. 

If,  now,  we  invert  the  plane  with  A  as  the  center  of  inversion, 
the  resultant  transformation,  1C,  must  be  a  similitude,  since  it 
transforms  lines  into  lines  and  circles  into  circles.  Consequently, 

1C  =  S. 

If  we  invert  again  with  the  same  inversion  we  have, 
IIC  =  IS. 

But  the  same  inversion,  performed  twice  in  succession,  does  not 
alter  the  plane,  since  an  inversion  is  involutoric.  Hence, 

C  =  IS, 

and  we  conclude  that  C  is  a  circular  transformation.  If  the  simili- 
tude S  is  the  identity,  that  is,  if  it  does  not  change  the  plane  at  all, 
then  C  is  an  inversion. 

Exercises 

1.  If  two  polarities  have  in  common  a  self-polar  triangle  ABC, 
where  A  is  an  actual  point  and  B  and  C  are  two  conjugate  imaginary 
points  on  the  ideal  line  defined  by  the  elliptic  involution  set  up  on  the 
ideal  line  by  any  circle  in  the  plane,  show  that  the  quadric  transforma- 
tion arising  from  the  two  polarities  is  a  circular  transformation. 

2.  A  plane  is  reflected  across  a  fixed  line  lying  in  it;  that  is,  each 
element  is  replaced  by  its  symmetrical  element  with  respect  to  this 
line.     The  plane  is  then  inverted  with  respect  to  a  fixed  circle  whose 
center  lies  on  the  fixed  line.     Show  that  the  resultant  transformation 
is  a  circular  transformation. 

3.  Invert  a  hyperbola  together  with  its  asymptotes.     Draw  the 
inverse  curve:  (a)  when  the  center  of  inversion  is  outside  the  hyper- 
bola; (6)  inside  the  hyperbola;  (c)  on  the  hyperbola;  (d)  at  the  center 
of  the  hyperbola. 

168.  General  Note. — The  most  general  quadric  transformation 
is  a  birational  point-transformation  between  two  planes  such  that, 
in  general,  the  lines  in  one  plane  correspond  to  conies  in  the  other. 
There  are  three  fundamental  points  in  each  plane.  Lines  passing 


POLARITIES  IN  A  PLANE  AND  IN  A  BUNDLE     209 

through  a  fundamental  point  in  one  plane  are  transformed  into 
lines  passing  through  a  fundamental  point  in  the  other. 

The  fundamental  points  in  either  plane  may  be  all  distinct,  two 
coincident,  or  all  three  coincident;  they  may  be  all  real  or  two  of 
them  conjugate  imaginary. 

When  the  planes  are  superposed,  the  fundamental  points  in  one 
plane  may  be  all  distinct  from  the  fundamental  points  in  the  other; 
or  the  planes  may  have  one,  two,  or  three  fundamental  points  in 
common. 

In  the  last  case  we  have  the  involutoric  transformations  with 
which  we  began  in  Art.  163. 

The  inversions  were  first  considered  by  Dandelin  (1822).  If 
the  points  of  a  sphere  are  projected  from  the  extremities  of  a  diam- 
eter, upon  a  plane  perpendicular  to  the  diameter,  corresponding 
points  in  the  plane  are  inverse  with  respect  to  the  circle  which  the 
plane  cuts  from  the  sphere.  In  this  way,  one  or  the  other  of  the 
two  inversions  is  obtained  according  as  the  plane  cuts  the  sphere 
in  a  real  circle  or  not. 

Some  traces  of  inversion  are  found  earlier  in  the  writings  of 
Vieta  (1600)  and  Fermat  (1679). 

The  general  theory  of  circular  transformations  is  due  to  Mobius 
(1852-55),  who  called  it  Kreis-Verwandschaft. 

This  theory  is  of  special  importance  in  the  theory  of  functions 
of  the  imaginary  variable  and  its  application  to  mathematical 
physics,  since  linear  transformations  of  the  imaginary  variable  are 
represented  geometrically  by  circular  transformations  in  a  real 
plane. 

The  inversions  were  the  first  quadric  transformations  discovered. 
Those  arising  from  two  polarities  were  discovered  by  Magnus 
(1832-33).  The  perspective  quadric  transformations  were  dis- 
covered by  Bellavitis  (1838). 

The  more  general  quadric  transformations  have  been  studied  by 
many  geometricians  and  from  various  points  of  view,  notably: 
Plucker  (1830-35),  Steiner  (1832),  and  Seydewitz  (1846). 

The  most  general  birational  point-transformations  between  two 
planes  were  discovered  by  Cremona  (1863-65)  and  are  called  after 
him,  Cremona  transformations.  The  quadric  transformations 
are  special  cases  of  Cremona  transformations. 

14 


INDEX 

(Numbers  refer  to  pages.) 


Affinity,  170 

corresponding  conies  in,  172 

fundamental  property  of,  170 
Algebraic  equations  of  conies,  100 
Anti-polarity,  190 
Apollonius,  36 

circle  of,  34 

theorem  of,  174 
Area,  of  parabolic  segment,  173 

of  ellipse,  175 

Assumption,  fundamental,  5 
Asymptotes,  66 
Asymptotic  cone,  110 
Axis,  of  a  cone,  200 

of  a  conic,  97 

of  a  cylinder,  104 

of  perspectivity,  22,  166 

B 

Bellavitis,  209 
Brianchon,  80 

theorem  of,  68 
Bundle  of  lines,  of  planes,  2 


Center,  of  conies,  94 

of  perspectivity,  22,  166 
of  projectivity,  118 
of  surfaces,  1 10 

Chains,  of  perspectivityj  37,  114,  160 


Chord  of  contact,  83 
Circle,  of  Apollonius,  34 

auxiliary,  156 

director,  158 
Circular  involution,  132 

transformations,  207 
Circumscribing  tetrahedrons,  109 
Classification,  of  conies,  66 

of  polarities,  187 

of  projectivities,  120 

of  surfaces,  109 
Coaxial  planes,  22 
Collinear  points,  22 
Collineation,  161 

limiting  lines  of,  168 

self -corresponding  elements,  179 
Complete,  n-edge,  13 

n-face,  13 

n-plane,  14 

n-point,  12,  14 

n-side,  13 

Concurrent  lines,  22 
Configuration,  of  Desargues,  15 

of  Pappus,  70 
Confocal,  elements,  196 

conies,  157 
Cones,  79,  199 

Conform  transformations,  206 
Congruence,  178 
Conic  sections,  66 
Conjugate,  of  an  ideal  point,  30 

diameters,  94 

diametral  planes,  104 

points  and  lines,  86 


211 


212 


INDEX 


Construction  of  confocal  elements, 
196 

of  curves  and  envelopes,  62 

of  cyclic  projectivities,  121 

of  foci,  146 

of  harmonic  ranges,  27 

of  perspectivity,  166 

of  polarity,  184 

of  poles  and  polar  lines,  84 
Continuity,  theorem  of,  43 

principle  of,  77 

Converse  theorems,  22,  40,  69,  122 
Coplanar  figures,  22 
Correlation  of  geometric  figures,  16 
Cremona,  209 
Cross-ratio,  31 

of  harmonic  range,  32 
Curves,  classification  of,  66 

of  second  order,  56 
Cyclic  planes,  201 

projectivities,  121 
Cylinders,  79 


D 


Dandelin,  209 
Dedekind  postulate,  42 
Desargues,  15 

configuration  of,  15,  22 
.  theorem  of,  20,  134 
Determination  of  projective  relation- 
ship, 50,  163 
Diagonal  pyramid,  92 

triangle,  91 
Diameters,  94 
Diametral  planes,  104 
Dimensions  of  primitive  forms,  19 
Directrices,  149 
Duality,  161 

in  a  plane,  8 

in  space,  9 

principle  of,  10 


E 


Elements,  the,  1 

common,  of  double  polarity,  193 

confocal,  of  double  polarity,  196 

double,  of  projectivity,  119 

double    conjugate,     of    double 
polarity,  194 

dual,  8 

fundamental,  202 

homologous,  16 

ideal,  5 

reciprocal,  8 

self-corresponding,  44,  116 

separated,  39,  40 
Elementary  forms,  55 

relations  among,  64 

totality  of,  66 
Envelope,  56 
Ellipse,  66 
Equation,  of  circle,  102 

of  ellipse,  100 

of  hyperbola,  100 

of  parabola,  103 


Fermat,  209 

Field  of  points,  of  lines,  2 

Figures,  complete,  12 

coplanar,  22 

geometric,  16 

simple,  10 

in  space,  13 
Focal  radii,  150 

axes,  201 
Focus,  146 
Forms,  elementary,  55 

primitive,  2 

G 

Generation  of  elementary  forms,  55 


INDEX 


213 


Generation  of  elementary  v  forms  of 
particular  conies  and  envelopes, 
78 

Generator  of  ruled  surface,   105 

Geometric  figures,  16 
mean,  33 

Gergonne,  15 

H 

Harmonic  elements,  112 

conjugates,  27 

mean,  32 

pencils,  28 

ranges,  25 

scales,  38 
Hyperbola,  66 
Hyperboloids,  110 


Imaginary  elements,  141 

lines,  138,  141 

planes,  139 

points,  137 
Infinitely  distant,  6 
Invariant  of  perspectivity,  167 
Inversion,  204 
Involution,  127,  168 

circular,  132 

determined  by  quadrangle,  133 

determined     by     quadrilateral, 
133 

elliptic,  128 

hyperbolic,  128 

on  a  sheaf  of  rays,  132 

on  a  straight  line,  130 

parabolic,  129 


Krois-Verwandshaft,  209 


Limiting  lines  of  a  collineation,  168 
Linear  complex,  special,  3 


M 


Magnus,  209 
Mobius,  54,  209 


X 


Normal  conjugate  rays,  30 
Notation,  2,  19 

O 

Order,  effect  of,  26 
Orthocenter,  137 
Orthogonal  circles,  34 
Orthogonally  correlated  bundles,  160 


Pappus,  36 

configuration  of,  70 
Parabola,  66 
Parallel,  5 
Pascal,  80 

theorem  of,  68 
Pencil,  harmonic,  28 
Pentagon  theorem,  73 
Perspective  position,  16,  113,  159 
Perspectivity,  165 

harmonic,  168 

invariant  of,  167 
Planes,  cyclic,  201 

diametral,  200 

of  symmetry,  104 

principal,  104 
Pliicker,  209 
Point  of  contact,  58 


214 


INDEX 


Point-row,  2 
Polar  figures,  89 

planes,  108 
Polarity,  absolute,  192 

double,  192 

in  a  bundle,  189 

in  a  plane,  183 

orthogonal,  190 

self-conjugate  elements  of,  185 
Poles  and  polar  lines,  81 

special  positions  of,  83 
Pole-rays  and  polar  planes,  91 
Poncelet,  15 

Postulate,  Dedekind,  42 
Primitive  forms,  of  the  first  kind,  37 

of  the  second  kind,  159 
Principle  of  duality,  10 

of  continuity,  77 
Projection,  central,  1 

and  section,  3 
Projective  relationship,  38,  114,  160 

determination  of,  50,  115,  163 
Projectivity,  38,  165 

cyclic,  121 

elliptic,  120 

hyperbolic,  120 

parabolic,  120 
Projector,  3 

of  regulus,  65 

Properties  of  curves  and  envelopes, 
57 

of  the  similitude,  176 

reflection,  154 


Q 


Quadrangle,  complete,  12 

theorem,  76 
Quadrangles  in  perspective  position, 

23 
Quadric  transformations,  202 

perspective,  203 


Quadrilateral,  complete,  13 
R 

Radical  axis,  177 
Ranges,  harmonic,  25 
Reflection  properties,  154 
Regulus,  57 

S 

Section,  3 

of  regulus,  65 

of  surface,  106 

Self-conjugate  elements,  185,  190 
Self-corresponding  elements,  44,  179 
Self -polar  figures,  91 
Seydewitz,  209 
Sheaf,  of  lines  or  planes,  2 

of  planes  of  second  class,  56 
Similarly  project  ive,  79 
Similitude,  176 

inverse  points,  177 . 

properties  of,  176 
Simple  n-edge,  13 

n-face,  13 

n-point,  11 

n-side,  11 
Space,  of  points,  of  rays,  of  planes, 

3 

Steiner,  93,  209 
Surfaces,  ruled,  105 

classification  of,  109 
Superposition,  44,  116 


Tangent  cones,  108 

lii»>,  58 

lines  and  planes.  107 
Theorem,  of  Apollonius.  1 7  1 

of  Brianchon,  68 


INDEX  215 

Theorem,  of  continuity,  43  Transformation,   Cremona,  209 

of  Desargues,  20,  134  quadric,  202 

of  Pascal,  68 

of  Von  Staudt,  46  V 

Theory  of  involution,  127 

Trace,  3  Vertices  of  a  conic,  99 

Transformation,  birational,  207  Vieta,  209 

circular,  207  Von  Staudt,  54 

conform,  206 


n.   Bowling.  Protective  geometry. 

SiA 

**•?!  UNIVERSITY  OF  CALIFORNIA  LIBRARY 

D?PP  Los  Angeles 

This  book  is  DUE  on  the  last  date  stamped  below. 


APR  3     1964 


JUN  1  2  1964 


REC'D  COL  LU 


MAR  16 


NOV  1 


JAN  3  1  1 


41988 


HA*  3  m 


MAR  1  0  RflTC 


31986 

Form  L9-116m-8,'62(D1237s8)444 


kj.  30-  II 

MAYS    1971 
[JW   X  X  1971 

JOL  14RECTI 


3  1918 


Engineering  & 


JUL72 


